Geometry 12 — 100-Question Mastery Exam

Q001/100 Foundations & Logic Set 1

Which of the following is an undefined term in geometry — a basic building block that is only described, not formally defined?

Undefined terms are the starting vocabulary every other definition is built from.

Concept: Undefined terms: point, line, plane.

Point, line, and plane are the three undefined terms in geometry. Every other term (angle, segment, triangle) is defined using them.

Q002/100 Foundations & Logic Set 1

If two distinct lines intersect, at how many points do they intersect?

Two straight, non-identical lines can cross only once.

Concept: Postulate: two lines intersect in at most one point.

Two distinct straight lines either never meet (parallel) or meet at exactly one point — they cannot cross twice.

Q003/100 Angles & Parallel Lines Set 1

Two angles are complementary, with measures 3x and 2x. What is x?

Complementary angles add to 90°.

Concept: Complementary angles sum to 90°.

3x + 2x = 90, so 5x = 90 and x = 18.

Q004/100 Angles & Parallel Lines Set 1

Vertical angles, formed when two lines cross, are always:

Think of an X shape — the two angles across from each other look identical.

Concept: Vertical Angles Theorem.

Vertical angles are the two angles opposite each other at an intersection, and they are always congruent.

Q005/100 Triangle Congruence Set 1

Which triangle congruence shortcut uses two angles and the included side between them?

The side must be sandwiched between the two marked angles.

Concept: ASA: Angle-Side-Angle.

ASA requires two angles and the side that lies directly between them to be congruent.

Q006/100 Triangle Congruence Set 1

Which of these is NOT a valid way to prove two triangles congruent?

This one only proves triangles are the same shape, not necessarily the same size.

Concept: AAA proves similarity, not congruence.

AAA (Angle-Angle-Angle) only guarantees the triangles are similar — same shape, possibly different size — so it never proves congruence.

Q007/100 Triangle Relationships Set 1

A triangle has two sides of length 5 and 7. Which value below could be the length of the third side?

The third side must be less than the sum and more than the difference of the other two.

Concept: Triangle Inequality Theorem: |a-b| < c < a+b.

The third side must satisfy 7−5 < x < 7+5, i.e. 2 < x < 12. Only 8 fits.

Q008/100 Similarity Set 1

Two polygons are similar when corresponding angles are congruent and corresponding sides are:

Similar figures keep the same shape but can change size by a consistent ratio.

Concept: Similarity = congruent angles + proportional sides.

Similar polygons have congruent corresponding angles, and their corresponding sides share one constant ratio (scale factor).

Q009/100 Right Triangles & Trigonometry Set 1

A right triangle has legs of 9 and 12. What is the length of the hypotenuse?

This is a scaled-up version of the classic 3-4-5 triangle.

Concept: Pythagorean Theorem: a² + b² = c².

9² + 12² = 81 + 144 = 225, and √225 = 15. (This is the 3-4-5 triple scaled by 3.)

Q010/100 Right Triangles & Trigonometry Set 1

In a right triangle, sin(θ) is defined as the ratio of which two sides?

SOH-CAH-TOA — the first letter tells you the ratio for sine.

Concept: sin(θ) = opposite / hypotenuse.

By definition, sin(θ) = length of the side opposite θ, divided by the hypotenuse.

Q011/100 Polygons & Quadrilaterals Set 1

What is the sum of the interior angles of a regular octagon (8 sides)?

Use (n − 2) × 180°.

Concept: Interior angle sum = (n − 2) × 180°.

(8 − 2) × 180° = 6 × 180° = 1080°.

Q012/100 Polygons & Quadrilaterals Set 1

What is the measure of one interior angle of a regular hexagon?

First find the total sum, then divide by the number of sides.

Concept: Interior angle sum ÷ n, for a regular polygon.

Sum = (6−2)×180° = 720°. Each angle = 720° ÷ 6 = 120°.

Q013/100 Circles Set 1

A central angle of a circle measures 80°. What is the measure of its intercepted arc?

A central angle and its arc always match.

Concept: Central angle = intercepted arc measure.

A central angle is, by definition, equal in measure to the arc it intercepts — both are 80°.

Q014/100 Circles Set 1

The diameter of a circle is always twice the length of the:

Cut the diameter exactly in half to get this segment.

Concept: Diameter = 2 × radius.

The radius runs from the center to the circle; the diameter runs all the way across through the center, so it equals 2r.

Q015/100 Coordinate Geometry Set 1

Find the distance between the points (1, 2) and (7, 10).

Build a right triangle using the horizontal and vertical differences.

Concept: Distance formula: d = √[(x₂−x₁)² + (y₂−y₁)²].

d = √[(7−1)² + (10−2)²] = √[36 + 64] = √100 = 10.

Q016/100 Transformations Set 1

Point (3, −5) is reflected over the x-axis. What are the coordinates of its image?

Reflecting over the x-axis flips the sign of only one coordinate.

Concept: Reflection over x-axis: (x, y) → (x, −y).

Only the y-coordinate changes sign: (3, −5) → (3, 5).

Q017/100 Transformations Set 1

A translation moves every point using the rule (x, y) → (x + 4, y − 2). Where does (1, 3) land?

Add the rule's values directly to the original coordinates.

Concept: Translation shifts every point by the same fixed amount.

(1 + 4, 3 − 2) = (5, 1).

Q018/100 Area & Perimeter Set 1

A triangle has a base of 10 units and a height of 6 units. What is its area?

Don't forget the one-half in the formula.

Concept: Area of a triangle = ½ × base × height.

Area = ½ × 10 × 6 = 30 square units.

Q019/100 Surface Area & Volume Set 1

A rectangular prism has length 5, width 4, and height 3. What is its volume?

Multiply all three dimensions together.

Concept: Volume of a rectangular prism = l × w × h.

V = 5 × 4 × 3 = 60 cubic units.

Q020/100 Surface Area & Volume Set 1

A cube has a side length of 5. What is its total surface area?

A cube has 6 identical square faces.

Concept: Surface area of a cube = 6s².

SA = 6 × 5² = 6 × 25 = 150 square units.

Q021/100 Probability Set 1

A fair six-sided die is rolled once. What is the probability of rolling an even number?

List the even outcomes out of all six possibilities.

Concept: Probability = favorable outcomes ÷ total outcomes.

Even outcomes are 2, 4, 6 — that's 3 out of 6, which simplifies to 1/2.

Q022/100 Foundations & Logic Set 1

What is the converse of the statement: "If it is raining, then the ground is wet"?

The converse swaps the hypothesis and conclusion.

Concept: Converse: switch the "if" and "then" parts.

The converse swaps hypothesis and conclusion: "If the ground is wet, then it is raining."

Q023/100 Foundations & Logic Set 1

A biconditional statement ("p if and only if q") is true exactly when:

Think of it as two conditionals that must agree with each other.

Concept: Biconditional: true only when both parts match (both T or both F).

A biconditional is true precisely when p and q share the same truth value — both true, or both false.

Q024/100 Angles & Parallel Lines Set 1

Two angles form a linear pair with measures (4x + 20)° and (2x + 40)°. Find x.

A linear pair always adds up to a straight line.

Concept: Linear pairs are supplementary — they sum to 180°.

(4x+20)+(2x+40) = 180 → 6x + 60 = 180 → 6x = 120 → x = 20.

Q025/100 Angles & Parallel Lines Set 1

Lines l and m are parallel and cut by transversal t. One interior angle measures 65°. What is the measure of its alternate interior angle?

Alternate interior angles are always the same size when lines are parallel.

Concept: Alternate Interior Angles Theorem (parallel lines).

When lines are parallel, alternate interior angles are congruent, so the alternate angle is also 65°.

Q026/100 Triangle Congruence Set 2

What does CPCTC stand for, and when is it used?

It's the last step used right after proving two triangles congruent.

Concept: CPCTC lets you conclude leftover parts are congruent once triangles are proven congruent.

CPCTC = Corresponding Parts of Congruent Triangles are Congruent. It's used after a congruence proof to justify that remaining sides/angles also match.

Q027/100 Triangle Congruence Set 2

The Hypotenuse-Leg (HL) congruence theorem can only be applied to:

The name of the theorem tells you exactly which side must exist.

Concept: HL applies only when both triangles have a right angle.

HL requires a hypotenuse, which only exists in a right triangle, so it applies to right triangles only.

Q028/100 Triangle Relationships Set 2

In a triangle, the centroid divides each median into two segments. What is the ratio, from the vertex to the centroid to the midpoint?

The centroid is closer to the midpoint of the side than to the vertex... or is it the other way around? Check the longer piece.

Concept: Centroid Theorem: divides each median 2:1 from the vertex.

The centroid always splits a median so the vertex-to-centroid piece is twice as long as the centroid-to-midpoint piece — a 2:1 ratio.

Q029/100 Triangle Relationships Set 2

A triangle's midsegment connects the midpoints of two sides. If the third (parallel) side of the triangle is 18 units, how long is the midsegment?

The midsegment is always half the side it's parallel to.

Concept: Triangle Midsegment Theorem: midsegment = ½ × parallel side.

Midsegment = ½ × 18 = 9 units.

Q030/100 Similarity Set 2

If two angles of one triangle are congruent to two angles of another triangle, the triangles are similar by:

Only two matching angles are needed — the third angle is automatically determined.

Concept: AA Similarity Postulate.

Since triangle angles sum to 180°, two congruent angle pairs force the third pair to match too, guaranteeing similarity (AA).

Q031/100 Similarity Set 2

Triangle ABC has sides 4, 6, 8. Triangle DEF has corresponding sides 6, 9, 12. What is the scale factor from ABC to DEF?

Divide a side of DEF by its corresponding side in ABC.

Concept: Scale factor = ratio of corresponding sides.

6/4 = 9/6 = 12/8 = 3/2, so the scale factor from ABC to DEF is 3/2.

Q032/100 Right Triangles & Trigonometry Set 2

A right triangle has a 45°-45°-90° angle set and a leg of length 9. What is the hypotenuse?

In this special triangle, the hypotenuse is always the leg times one specific radical.

Concept: 45-45-90 triangle: hypotenuse = leg × √2.

Hypotenuse = 9√2.

Q033/100 Right Triangles & Trigonometry Set 2

A 30°-60°-90° triangle has a hypotenuse of 12. What is the length of the longer leg (opposite the 60° angle)?

First find the short leg (half the hypotenuse), then multiply by √3.

Concept: 30-60-90 triangle: short leg = hyp/2, long leg = short leg × √3.

Short leg = 12/2 = 6. Long leg = 6√3.

Q034/100 Polygons & Quadrilaterals Set 2

What is the sum of the exterior angles (one at each vertex) of any convex polygon?

This total never changes, no matter how many sides the polygon has.

Concept: Exterior Angle Sum Theorem: always 360° for any convex polygon.

The exterior angles of any convex polygon, one per vertex, always sum to exactly 360°.

Q035/100 Polygons & Quadrilaterals Set 2

In a parallelogram, the two diagonals always:

Think about where the diagonals cross.

Concept: Parallelogram property: diagonals bisect each other.

In every parallelogram, the diagonals cut each other exactly in half at their point of intersection.

Q036/100 Circles Set 2

An inscribed angle intercepts an arc of 140°. What is the measure of the inscribed angle?

Inscribed angles are always "half" of something.

Concept: Inscribed Angle Theorem: inscribed angle = ½ × intercepted arc.

Inscribed angle = 140° ÷ 2 = 70°.

Q037/100 Circles Set 2

A tangent line to a circle is always _____ to the radius drawn to the point of tangency.

Picture a wheel touching a flat road — what angle does the spoke make with the road?

Concept: Tangent–Radius Theorem.

A tangent line meets the radius at the point of tangency at exactly 90° — they are always perpendicular.

Q038/100 Coordinate Geometry Set 2

Find the midpoint of the segment connecting (1, 2) and (7, 10).

Average the x-coordinates, then average the y-coordinates.

Concept: Midpoint formula: ((x₁+x₂)/2, (y₁+y₂)/2).

Midpoint = ((1+7)/2, (2+10)/2) = (4, 6).

Q039/100 Coordinate Geometry Set 2

What is the slope of the line through (2, 3) and (5, 9)?

Slope is rise over run — the change in y divided by the change in x.

Concept: Slope formula: m = (y₂−y₁)/(x₂−x₁).

m = (9−3)/(5−2) = 6/3 = 2.

Q040/100 Transformations Set 2

A point (3, 5) is rotated 90° counterclockwise about the origin. What is its image?

The rule for a 90° CCW rotation swaps the coordinates and flips one sign.

Concept: 90° CCW rotation about the origin: (x, y) → (−y, x).

(3, 5) → (−5, 3).

Q041/100 Transformations Set 2

A dilation with scale factor 2, centered at the origin, is applied to point (4, −2). What is the image?

Multiply both coordinates by the scale factor.

Concept: Dilation about the origin: (x, y) → (kx, ky).

(4×2, −2×2) = (8, −4).

Q042/100 Area & Perimeter Set 2

A trapezoid has parallel bases of 8 and 12 units, and a height of 5 units. What is its area?

Average the two bases first, then multiply by the height.

Concept: Area of a trapezoid = ½(b₁ + b₂) × h.

Area = ½(8+12) × 5 = ½(20)(5) = 50 square units.

Q043/100 Surface Area & Volume Set 2

A cylinder has radius 4 and height 10. What is its volume, in terms of π?

Multiply the base circle's area by the height.

Concept: Volume of a cylinder = πr²h.

V = π(4²)(10) = 160π.

Q044/100 Surface Area & Volume Set 2

A cone has radius 6 and height 8. What is its volume, in terms of π?

A cone holds exactly one-third the volume of a cylinder with the same base and height.

Concept: Volume of a cone = (1/3)πr²h.

V = (1/3)π(6²)(8) = (1/3)π(288) = 96π.

Q045/100 Probability Set 2

A point is chosen at random on a 20-unit segment. What is the probability it lands within a specific 5-unit sub-segment?

This is a length-based (geometric) probability — favorable length over total length.

Concept: Geometric probability = favorable measure ÷ total measure.

P = 5/20 = 1/4.

Q046/100 Probability Set 2

A square dartboard has an inscribed circle touching all four sides. If the square has side 10 (so the circle has radius 5), what is the probability a random dart lands inside the circle?

Compare the circle's area to the square's area.

Concept: Geometric probability with area = circle area ÷ square area.

Circle area = π(5²) = 25π. Square area = 10² = 100. P = 25π/100 = π/4.

Q047/100 Foundations & Logic Set 2

Given: "If a shape is a square, it is a rectangle" and "If a shape is a rectangle, it is a parallelogram." By the Law of Syllogism, what can you conclude?

Chain the two conditionals together like dominoes.

Concept: Law of Syllogism: if p→q and q→r, then p→r.

Square→rectangle and rectangle→parallelogram chain together to give: square→parallelogram.

Q048/100 Angles & Parallel Lines Set 2

A same-side (co-interior) interior angle pair is formed by parallel lines and a transversal. One angle measures 72°. What is the other?

This pair doesn't match — it adds up to a straight angle instead.

Concept: Same-Side Interior Angles Theorem: the pair is supplementary.

Same-side interior angles sum to 180°, so the other angle is 180° − 72° = 108°.

Q049/100 Angles & Parallel Lines Set 2

In a triangle, one exterior angle equals the sum of the two remote (non-adjacent) interior angles. If those two interior angles are 50° and 65°, what is the exterior angle?

Add the two "far away" interior angles together.

Concept: Triangle Exterior Angle Theorem.

Exterior angle = 50° + 65° = 115°.

Q050/100 Triangle Congruence Set 2

In an isosceles triangle, the two angles opposite the congruent sides are always:

This is the reason isosceles triangles look symmetric.

Concept: Isosceles Triangle Theorem (Base Angles Theorem).

The base angles — opposite the two congruent sides — are always congruent to each other.

Q051/100 Triangle Congruence Set 3

Triangles ABC and DEF have AB = DE, ∠B ≅ ∠E, and BC = EF. Which postulate proves the triangles are congruent?

Check where the congruent angle sits relative to the two sides.

Concept: SAS: the angle must be included between the two given sides.

∠B is between sides AB and BC (and ∠E is between DE and EF), so this is Side-Angle-Side (SAS).

Q052/100 Triangle Relationships Set 3

Which point of concurrency in a triangle is always equidistant from all three vertices?

This point is found using perpendicular bisectors of the sides.

Concept: Circumcenter: intersection of perpendicular bisectors; equidistant from vertices.

The circumcenter, formed where the three perpendicular bisectors meet, is equidistant from all three vertices (it's the center of the circumscribed circle).

Q053/100 Triangle Relationships Set 3

Which point of concurrency in a triangle is always equidistant from all three sides?

This point is found using the angle bisectors.

Concept: Incenter: intersection of angle bisectors; equidistant from sides.

The incenter, formed where the three angle bisectors meet, is equidistant from all three sides (it's the center of the inscribed circle).

Q054/100 Similarity Set 3

A larger triangle has a perimeter of 40, and its scale factor to a smaller similar triangle is 5:3 (larger:smaller). What is the smaller triangle's perimeter?

The perimeter ratio matches the side-length scale factor directly.

Concept: Similar figures: perimeter ratio = scale factor.

Smaller perimeter = 40 × (3/5) = 24.

Q055/100 Similarity Set 3

Two similar triangles have a scale factor of 2:3 (smaller:larger). If the larger triangle's area is 54, what is the smaller triangle's area?

Area ratio is the scale factor squared, not the scale factor itself.

Concept: Similar figures: area ratio = (scale factor)².

Area ratio = (2/3)² = 4/9. Smaller area = 54 × 4/9 = 24.

Q056/100 Right Triangles & Trigonometry Set 3

A right triangle has a 60° angle with an adjacent side of length 5. What is the length of the side opposite the 60° angle?

Use the tangent ratio: opposite over adjacent.

Concept: tan(60°) = opposite/adjacent, and tan(60°) = √3.

opposite = 5 × tan(60°) = 5√3.

Q057/100 Right Triangles & Trigonometry Set 3

A ramp rises to a height h over a horizontal distance of 20 units, at a 45° angle of elevation. What is h?

At exactly 45°, the rise and the run are always equal.

Concept: tan(angle of elevation) = height / horizontal distance.

tan(45°) = 1 = h/20, so h = 20.

Q058/100 Polygons & Quadrilaterals Set 3

The diagonals of a rhombus are always:

A rhombus's diagonals cross at a right angle, cutting each other in half.

Concept: Rhombus property: diagonals are perpendicular bisectors of each other.

In a rhombus, the diagonals always bisect each other AND meet at 90°.

Q059/100 Polygons & Quadrilaterals Set 3

A trapezoid has bases of 10 and 16. How long is its midsegment (the segment joining the midpoints of the two legs)?

Average the two bases.

Concept: Trapezoid Midsegment Theorem: midsegment = ½(b₁ + b₂).

Midsegment = ½(10 + 16) = ½(26) = 13.

Q060/100 Circles Set 3

Two chords intersect inside a circle. One chord is split into segments of 4 and 9; the other is split into segments of 6 and x. Find x.

The products of the two pairs of segments must be equal.

Concept: Intersecting Chords Theorem: product of segments of one chord = product of segments of the other.

4 × 9 = 6 × x → 36 = 6x → x = 6.

Q061/100 Circles Set 3

On a circle, arc AB = 110° and arc BC = 95°, where B lies between A and C. What is the measure of arc AC (passing through B)?

Add the two smaller arcs together.

Concept: Arc Addition Postulate.

Arc AC (through B) = arc AB + arc BC = 110° + 95° = 205°.

Q062/100 Coordinate Geometry Set 3

Given a slope of 2/3, what is the slope of a line perpendicular to it?

Perpendicular slopes are negative reciprocals of each other.

Concept: Perpendicular lines: slopes multiply to −1.

The negative reciprocal of 2/3 is −3/2. Check: (2/3)(−3/2) = −1. ✓

Q063/100 Coordinate Geometry Set 3

What is the equation of the line through point (2, 3) with slope 2?

Plug the point into point-slope form and simplify.

Concept: Point-slope form: y − y₁ = m(x − x₁).

y − 3 = 2(x − 2) → y = 2x − 4 + 3 → y = 2x − 1. Check: at x=2, y=3. ✓

Q064/100 Transformations Set 3

Reflecting a figure over two parallel lines, one after the other, produces the same result as a single:

The figure ends up facing the same way it started, just shifted.

Concept: Composition Theorem: two reflections over parallel lines = one translation.

A composition of reflections over two parallel lines is always equivalent to a single translation, perpendicular to the lines, by twice the distance between them.

Q065/100 Area & Perimeter Set 3

What is the area of a regular hexagon with side length 6?

A regular hexagon splits into 6 equilateral triangles.

Concept: Area of a regular hexagon = (3√3/2)s².

Area = (3√3/2)(6²) = (3√3/2)(36) = 54√3 square units.

Q066/100 Area & Perimeter Set 3

A circle has a radius of 7. What is its area, in terms of π?

Square the radius, then multiply by π.

Concept: Area of a circle = πr².

Area = π(7²) = 49π.

Q067/100 Surface Area & Volume Set 3

A sphere has a radius of 9. What is its volume, in terms of π?

Cube the radius, then apply the sphere volume formula.

Concept: Volume of a sphere = (4/3)πr³.

V = (4/3)π(9³) = (4/3)π(729) = 972π.

Q068/100 Surface Area & Volume Set 3

A sphere has a radius of 7. What is its total surface area, in terms of π?

Square the radius, then multiply by 4π.

Concept: Surface area of a sphere = 4πr².

SA = 4π(7²) = 4π(49) = 196π.

Q069/100 Probability Set 3

Events A and B are independent, with P(A) = 1/3 and P(B) = 1/4. What is P(A and B)?

For independent events, multiply the individual probabilities.

Concept: P(A and B) = P(A) × P(B), for independent events.

P(A and B) = (1/3)(1/4) = 1/12.

Q070/100 Angles & Parallel Lines Set 3

Two lines are cut by a transversal, and a pair of alternate exterior angles are congruent. What must be true about the two lines?

This is the reverse of the theorem you already know — it proves lines are parallel.

Concept: Converse of the Alternate Exterior Angles Theorem.

If alternate exterior angles are congruent, the lines that the transversal cuts must be parallel — this is a converse theorem used to prove lines parallel.

Q071/100 Triangle Congruence Set 3

In a proof, two triangles share side BD. It's given that ∠ABD ≅ ∠CDB and AB ≅ CD, and BD ≅ BD by the Reflexive Property. Which postulate proves △ABD ≅ △CDB?

Find the two sides and check whether the given angle sits between them.

Concept: SAS with a shared (reflexive) side.

AB ≅ CD (given), ∠ABD ≅ ∠CDB (given, included between the sides), BD ≅ BD (reflexive) → SAS.

Q072/100 Triangle Congruence Set 3

Two triangles have two pairs of congruent angles, and a pair of congruent sides that is NOT between those angles. Which theorem proves congruence?

The side is congruent, but it sits outside the two marked angles.

Concept: AAS: Angle-Angle-Side.

When the congruent side is not included between the two congruent angles, the correct theorem is AAS.

Q073/100 Triangle Relationships Set 3

In an obtuse triangle, where is the orthocenter (the intersection of the three altitudes) located?

Think about what happens to an altitude's line when a triangle stretches wide open.

Concept: The orthocenter's location depends on the triangle's angle type.

For an obtuse triangle, at least one altitude must be extended outside the triangle to meet the others, so the orthocenter lies outside the triangle.

Q074/100 Similarity Set 3

In similar triangles, x/8 = (x+3)/12. What is the value of x?

Cross-multiply to clear the fractions.

Concept: Solving a proportion from similar triangles by cross-multiplication.

12x = 8(x+3) → 12x = 8x + 24 → 4x = 24 → x = 6.

Q075/100 Right Triangles & Trigonometry Set 3

Using the Law of Sines, a triangle has ∠A = 30°, side a = 6, and ∠B = 60°. Find side b.

Set up the Law of Sines ratio and solve — this pairs with a familiar special-triangle radical.

Concept: Law of Sines: a/sin A = b/sin B.

6/sin(30°) = b/sin(60°) → 6/0.5 = b/(√3/2) → 12 = b/(√3/2) → b = 12 × (√3/2) = 6√3.

Q076/100 Right Triangles & Trigonometry Set 4

Using the Law of Cosines, a triangle has sides a = 5, b = 8, and the included angle C = 60°. Find side c.

Plug directly into the Law of Cosines formula and simplify carefully.

Concept: Law of Cosines: c² = a² + b² − 2ab·cos(C).

c² = 25 + 64 − 2(5)(8)(0.5) = 89 − 40 = 49, so c = 7.

Q077/100 Polygons & Quadrilaterals Set 4

Which special quadrilateral always has exactly one diagonal that bisects the other (but not vice versa), and one pair of perpendicular diagonals?

This shape looks like the toy you fly on a windy day.

Concept: Kite property: perpendicular diagonals, only one bisects the other.

A kite has perpendicular diagonals where only the diagonal connecting the vertex angles is bisected by the other.

Q078/100 Polygons & Quadrilaterals Set 4

A regular polygon has an exterior angle of 24°. How many sides does it have?

The exterior angles of a regular polygon are all equal and must total 360°.

Concept: Number of sides = 360° ÷ exterior angle.

n = 360° ÷ 24° = 15.

Q079/100 Circles Set 4

From an external point, one secant has an external segment of 4 and a whole length of 12. A second secant from the same point has an external segment of 6. What is the length of the far segment (beyond the circle) of the second secant?

Set the two products of (external segment × whole length) equal to each other.

Concept: Secant-Secant Power Theorem: (ext₁)(whole₁) = (ext₂)(whole₂).

4 × 12 = 48. For the second secant, 6 × whole₂ = 48 → whole₂ = 8. Far segment = 8 − 6 = 2.

Q080/100 Circles Set 4

A tangent segment from an external point is 6 units. A secant from the same point has an external segment of 4 units. What is the length of the far segment of the secant, beyond the circle?

The tangent squared equals the secant's external segment times its whole length.

Concept: Tangent-Secant Power Theorem: tangent² = (external segment)(whole secant).

6² = 4 × whole → 36 = 4 × whole → whole = 9. Far segment = 9 − 4 = 5.

Q081/100 Coordinate Geometry Set 4

Which coordinate test most efficiently proves that a quadrilateral is a parallelogram?

One quick calculation on both diagonals settles it in a single step.

Concept: If the diagonals of a quadrilateral bisect each other, it must be a parallelogram.

Finding that both diagonals share the same midpoint proves they bisect each other, which is sufficient to prove a parallelogram — often faster than checking all four sides or angles.

Q082/100 Coordinate Geometry Set 4

A triangle has vertices A(0, 0), B(4, 0), and C(4, 3). What type of triangle is it?

Compare the slopes of the two sides that meet at B.

Concept: Perpendicular slopes (one horizontal, one vertical) indicate a right angle.

AB is horizontal (slope 0) and BC is vertical (undefined slope), so AB ⊥ BC, making this a right triangle with the right angle at B.

Q083/100 Transformations Set 4

Which of the following transformations is NOT an isometry (does not preserve size)?

Three of these four keep the figure exactly the same size — one changes it.

Concept: Isometries preserve distance; dilations only preserve shape (they resize).

Translations, reflections, and rotations all preserve size and shape. A dilation changes size (unless the scale factor is 1), so it is not an isometry.

Q084/100 Area & Perimeter Set 4

A 10-by-6 rectangle has a semicircular notch of radius 3 removed from one side (diameter 6 along that side). What is the remaining area?

Subtract the semicircle's area — not the full circle's — from the rectangle.

Concept: Composite area = rectangle area − semicircle area.

Rectangle area = 60. Semicircle area = ½π(3²) = 4.5π. Remaining area = 60 − 4.5π.

Q085/100 Surface Area & Volume Set 4

A silo-like solid is made of a cylinder (radius 5, height 12) topped with a cone (same radius, height 9). What is the total volume, in terms of π?

Find each solid's volume separately, then add them.

Concept: Total volume = cylinder volume + cone volume.

Cylinder: π(5²)(12) = 300π. Cone: (1/3)π(5²)(9) = 75π. Total = 300π + 75π = 375π.

Q086/100 Surface Area & Volume Set 4

Two similar solids have a scale factor of 1:3. If the smaller solid's volume is 8, what is the larger solid's volume?

Volume ratio is the scale factor cubed, not squared.

Concept: Similar solids: volume ratio = (scale factor)³.

Volume ratio = 3³ = 27. Larger volume = 8 × 27 = 216.

Q087/100 Probability Set 4

A dartboard is guaranteed to be hit somewhere inside a square dartboard of side 10 units. A circle of radius 5 is inscribed. What is the probability the dart does NOT land in the circle?

Find P(inside the circle) first, then subtract from 1.

Concept: Complement Rule: P(not A) = 1 − P(A).

P(inside circle) = 25π/100 = π/4. P(not inside) = 1 − π/4.

Q088/100 Triangle Congruence Set 4

In a two-column proof, you are given AB ≅ DE, ∠A ≅ ∠D, and AC ≅ DF. Which postulate completes the proof that △ABC ≅ △DEF?

Identify which side sits directly between the two triangles' congruent sides — check the angle's position.

Concept: SAS: the congruent angle must be included between the two congruent sides.

∠A is included between sides AB and AC (matching ∠D between DE and DF), so this is SAS.

Q089/100 Triangle Relationships Set 4

A triangle has sides of 9 and 14, and a third side of length x. Which of the following could be the triangle's perimeter?

First find the valid range for x, then add 9 + 14 to find the perimeter range.

Concept: Triangle Inequality Theorem applied to find a perimeter range.

5 < x < 23 (since 14−9 < x < 14+9). So perimeter = 23+x is between 28 and 46. Only 30 falls in that open range.

Q090/100 Similarity Set 4

Two similar triangles have side ratios producing the proportion x/(x+2) = 3/5, where a triangle's shortest side is x and the corresponding side of a similar triangle is x+2, scaled 3:5. What is x?

Cross-multiply and solve the linear equation carefully.

Concept: Cross-multiplying a similarity proportion.

5x = 3(x+2) → 5x = 3x + 6 → 2x = 6 → x = 3. (Re-check: shortest side x=3, other side x+2=5, ratio 3:5 ✓ — closest match is x = 3.)

Q091/100 Right Triangles & Trigonometry Set 4

A triangle has side lengths 5, 12, and 13. What is the measure of its largest angle?

Check whether these three lengths satisfy the Pythagorean Theorem before reaching for a calculator.

Concept: Converse of the Pythagorean Theorem: if a² + b² = c², the triangle is right.

5² + 12² = 25 + 144 = 169 = 13², so the triangle is right, and its largest angle (opposite the longest side) is 90°.

Q092/100 Right Triangles & Trigonometry Set 4

A triangle has sides a = 8, b = 10, and the included angle C = 30°. What is its area?

Use the SAS area formula — no height measurement needed.

Concept: Area = ½ab·sin(C).

Area = ½(8)(10)sin(30°) = ½(80)(0.5) = 20.

Q093/100 Circles Set 4

What is the standard-form equation of a circle with center (3, −2) and radius 5?

Remember the signs inside the parentheses are opposite the center's coordinates.

Concept: Standard form of a circle: (x−h)² + (y−k)² = r².

With center (3,−2) and r=5: (x−3)² + (y−(−2))² = 5² → (x−3)² + (y+2)² = 25.

Q094/100 Circles Set 4

A circle has radius 6. What is the area of a sector with a central angle of 60°, in terms of π?

Find what fraction of the full circle the sector represents.

Concept: Sector area = (central angle / 360°) × πr².

Sector area = (60/360) × π(6²) = (1/6)(36π) = 6π.

Q095/100 Circles Set 4

A circle has radius 9. What is the arc length for a central angle of 40°, in terms of π?

Find what fraction of the full circumference the arc represents.

Concept: Arc length = (central angle / 360°) × 2πr.

Arc length = (40/360) × 2π(9) = (1/9)(18π) = 2π.

Q096/100 Coordinate Geometry Set 4

Point P divides segment AB, where A(1, 1) and B(9, 5), in the ratio 3:1 from A to B. What are the coordinates of P?

P is 3/4 of the way from A to B — apply that fraction to both the horizontal and vertical change.

Concept: Section formula: P = A + (ratio fraction)(B − A).

P = (1 + ¾(9−1), 1 + ¾(5−1)) = (1 + 6, 1 + 3) = (7, 4).

Q097/100 Transformations Set 4

A point (2, 3) is first reflected over the y-axis, then translated using the rule (x, y) → (x+1, y−4). What is the final image?

Perform the reflection first, then apply the translation to that new point — order matters.

Concept: Composition of transformations: apply each transformation in the given order.

Reflect (2,3) over the y-axis: (−2,3). Translate: (−2+1, 3−4) = (−1, −1).

Q098/100 Area & Perimeter Set 4

A regular hexagon has side length 8. Its apothem is 4√3. What is its area?

Use the apothem-based area formula: half the perimeter times the apothem.

Concept: Area of a regular polygon = ½ × perimeter × apothem.

Perimeter = 6(8) = 48. Area = ½(48)(4√3) = 96√3.

Q099/100 Surface Area & Volume Set 4

Two similar solids have a scale factor of 2:5. If the smaller solid's surface area is 36, what is the larger solid's surface area?

Surface area ratio is the scale factor squared.

Concept: Similar solids: surface area ratio = (scale factor)².

Area ratio = (5/2)² = 25/4. Larger SA = 36 × 25/4 = 225.

Q100/100 Probability Set 4

A square dartboard has side length 10. A quarter-circle of radius 3, centered at one corner, marks a bonus zone. What is the probability a randomly thrown dart lands in the bonus zone?

Find the quarter-circle's area first, then divide by the full square's area.

Concept: Geometric probability with a quarter-circle region.

Quarter-circle area = ¼π(3²) = 2.25π. Square area = 100. P = 2.25π/100 = 9π/400.

The Geometry Mastery Exam

What's covered

Foundations & Logic

Angles & Parallel Lines

Triangle Congruence

Triangle Relationships

Similarity

Right Triangles & Trigonometry

Polygons & Quadrilaterals

Circles

Coordinate Geometry

Transformations

Area & Perimeter

Surface Area & Volume

Probability

Set your timer

Set 1 · Foundation Sweep

Set 2 · Core Practice

Set 3 · Applied Reasoning

Set 4 · Mastery Challenge

Score

Answer Key — Geometry 12 Mastery Exam