Algebra 1 · Geometry · Self-Study

Core Problem Set

20 essential problems — linear functions, inequalities, triangles, polygons & circles

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Algebra 1 — Linear Functions & Inequalities

Q 1–10
Q 01 Algebra Medium

Which equation represents a line with slope −3 passing through the point (2, 4)?

⚡ POINT-SLOPE: y − y₁ = m(x − x₁) → plug in and simplify to slope-intercept form y = mx + b
📖 Explanation
Use point-slope form: y − 4 = −3(x − 2) → y − 4 = −3x + 6 → y = −3x + 10.
Common mistake: students forget to distribute the negative sign → getting +6 (not −6) is the trap!
Q 02 Algebra Tricky

What is the x-intercept of the line 4x − 2y = 12?

⚡ X-INTERCEPT: set y = 0 and solve for x.  Y-INTERCEPT: set x = 0 and solve for y.
📖 Explanation
Set y = 0: 4x − 0 = 12 → 4x = 12 → x = 3. So x-intercept = (3, 0).
Choice A (6, 0) is wrong — students sometimes divide 12 by 2 instead of 4. Always use the x-coefficient!
Q 03 Algebra Medium

Lines ℓ₁ and ℓ₂ are perpendicular. If ℓ₁ has slope 2/3, what is the slope of ℓ₂?

⚡ PERPENDICULAR SLOPES: Flip & flip sign → m₂ = −1/m₁ (negative reciprocal)
📖 Explanation
Perpendicular slope = negative reciprocal of 2/3 = −3/2.
Choice C (−2/3) only flips the sign but doesn't flip the fraction — a very common error!
Q 04 Algebra Tricky

Solve the compound inequality: −2 ≤ 3x + 1 ≤ 10

−2 ≤ 3x + 1 ≤ 10
⚡ COMPOUND INEQUALITY: Do the same operation to ALL three parts simultaneously
📖 Explanation
Step 1 — subtract 1 from all parts: −3 ≤ 3x ≤ 9
Step 2 — divide all by 3: −1 ≤ x ≤ 3
Keep ≤ (closed), since original inequalities are also ≤. Choice B uses open inequalities — wrong!
Q 05 Algebra Medium

A linear function passes through (0, −5) and (4, 7). What is the rate of change (slope)?

Slope Formula
m = (y₂ − y₁) / (x₂ − x₁)  →  "Rise over Run"
📖 Explanation
m = (7 − (−5)) / (4 − 0) = 12 / 4 = 3
Note: 7 − (−5) = 7 + 5 = 12, not 2. Subtracting a negative is the #1 arithmetic trap here!
Q 06 Algebra Tricky

Which inequality describes the graph of a dashed line with slope 1 and y-intercept −2, with the region above shaded?

⚡ SHADING RULE: Above line → y > or y ≥  |  Below line → y < or y ≤  |  Dashed line → strict (< or >)
📖 Explanation
Dashed line → strict inequality (no equal). Above → y > ...
Line equation: y = x − 2, so answer = y > x − 2.
Choice A uses ≥ which means solid line — wrong because the line is dashed!
Q 07 Algebra Medium

If f(x) = −2x + 7, what is the value of f(−3)?

⚡ FUNCTION NOTATION: f(−3) means substitute x = −3 into the expression. Watch signs!
📖 Explanation
f(−3) = −2(−3) + 7 = +6 + 7 = 13
−2 × −3 = +6 (negative × negative = positive). Forgetting this gives B (1) — very common mistake!
Q 08 Algebra Tricky

Solve: |2x − 4| > 6

|2x − 4| > 6
⚡ ABSOLUTE VALUE INEQUALITY: |A| > k → A > k OR A < −k  (splits into TWO cases)
📖 Explanation
Case 1: 2x − 4 > 6 → 2x > 10 → x > 5
Case 2: 2x − 4 < −6 → 2x < −2 → x < −1
Answer: x < −1 or x > 5. Choice A describes |...| < 6 (the AND case) — a classic mix-up!
Q 09 Algebra Medium

Two lines are parallel. Line 1: y = 4x − 3. Which is Line 2?

⚡ PARALLEL LINES: Same slope, different y-intercept → never intersect
📖 Explanation
Rewrite B: 8x − 2y = 5 → 2y = 8x − 5 → y = 4x − 5/2. Slope = 4 ✓, y-intercept ≠ −3 ✓.
C has the exact same equation → same line (not parallel). A has slope −1/4 (perpendicular). D has slope 3.
Q 10 Algebra Tricky

When solving −4x ≤ 20, what is the correct answer?

−4x ≤ 20
⚡ FLIP THE SIGN! Dividing or multiplying both sides by a NEGATIVE number → reverse the inequality symbol
📖 Explanation
Divide both sides by −4 → FLIP ≤ to ≥: x ≥ 20/(−4) = x ≥ −5
Choice A forgets to flip the sign — the single most common inequality mistake in all of Algebra 1!
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Geometry — Triangles, Polygons & Circles

Q 11–20
Q 11 Geometry Medium

In a triangle, two angles measure 47° and 83°. What is the third angle?

⚡ TRIANGLE ANGLE SUM: Always 180°. Just subtract: 180 − (angle1 + angle2)
📖 Explanation
Third angle = 180° − 47° − 83° = 180° − 130° = 50°.
D (130°) is the sum of the two known angles — students sometimes think that IS the third angle!
Q 12 Geometry Tricky

In a right triangle with legs a = 6 and b = 8, what is the hypotenuse?

Pythagorean Theorem
a² + b² = c²  →  legs² + legs² = hyp²
📖 Explanation
6² + 8² = 36 + 64 = 100 → c = √100 = 10.
B (14) = 6 + 8 — students add the legs directly instead of using squares. Never add — use Pythagorean theorem!
Q 13 Geometry Medium

What is the sum of interior angles of a hexagon?

Interior Angle Sum Formula
S = (n − 2) × 180°    n = number of sides
📖 Explanation
Hexagon: n = 6. Sum = (6 − 2) × 180° = 4 × 180° = 720°.
A (540°) is for pentagon (n=5). D (1080°) is for octagon (n=8). Don't mix up n — count sides carefully!
Q 14 Geometry Tricky

A circle has radius 7 cm. What is its circumference? (Use π ≈ 3.14)

⚡ C vs A: Circumference = 2πr (perimeter)  |  Area = πr² (inside space)
📖 Explanation
C = 2πr = 2 × 3.14 × 7 = 43.96 cm.
A (153.86) = πr² (area formula — wrong). C has the right number but wrong unit (cm² is for area, not circumference)!
Q 15 Geometry Medium

An exterior angle of a triangle measures 115°. One of the non-adjacent interior angles is 60°. What is the other non-adjacent interior angle?

⚡ EXTERIOR ANGLE THEOREM: Exterior angle = sum of the TWO non-adjacent interior angles
📖 Explanation
Exterior angle = sum of remote interior angles: 115° = 60° + x → x = 55°.
A (65°) comes from subtracting from 180° instead of 115° — wrong theorem applied!
Q 16 Geometry Tricky

A regular polygon has each interior angle equal to 135°. How many sides does it have?

Each Interior Angle (Regular Polygon)
Each angle = (n − 2) × 180° / n  →  solve for n
📖 Explanation
135n = (n − 2) × 180 → 135n = 180n − 360 → 45n = 360 → n = 8 (octagon).
You can also use: exterior angle = 180 − 135 = 45°, and 360° / 45° = 8. Much faster!
Q 17 Geometry Medium

A circle has diameter 10 cm. What is its area? (Use π ≈ 3.14)

⚡ DIAMETER → RADIUS: r = d / 2. Always halve the diameter FIRST before using A = πr²
📖 Explanation
r = 10/2 = 5. Area = π × 5² = 3.14 × 25 = 78.5 cm².
B (314) uses r = 10 instead of r = 5 — forgetting to halve the diameter is the #1 circle mistake!
Q 18 Geometry Tricky

In △ABC and △DEF, AB/DE = BC/EF = AC/DF. What is the relationship between the two triangles?

⚡ CONGRUENT vs SIMILAR: Congruent (≅) = same shape & size. Similar (~) = same shape, different size
📖 Explanation
Three pairs of proportional sides (SSS~) → Similar triangles.
Congruent requires equal sides (ratio = 1). Proportional ratios only guarantee similarity, not congruence!
Q 19 Geometry Medium

A chord of a circle is 8 cm long. The radius of the circle is 5 cm. How far is the chord from the center?

Chord Distance
Draw a perpendicular from center to chord → it bisects the chord → use Pythagorean theorem
📖 Explanation
Perpendicular bisects chord → half-chord = 4 cm. Radius = 5 cm (hypotenuse).
d² + 4² = 5² → d² + 16 = 25 → d² = 9 → d = 3 cm. Recognize the 3-4-5 right triangle!
Q 20 Geometry Tricky

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

⚡ INSCRIBED ANGLE THEOREM: Inscribed angle = ½ × intercepted arc  |  Central angle = arc (same as arc)
📖 Explanation
Inscribed angle = ½ × arc = ½ × 140° = 70°.
A (140°) confuses inscribed angle with central angle (which DOES equal the arc). Remember: inscribed = half!
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