Self-Study Worksheet Β· Algebra 1 & Geometry

Master the Essentials

20 carefully crafted problems β€” multiple choice with instant feedback and full explanations.

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Part 1

Algebra 1

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QUICK MEMORY POINTS ISOLATE β†’ Move everything to one side, variable alone.  |  DISTRIBUTE β†’ Multiply inside parentheses first.  |  SLOPE = rise/run = (yβ‚‚βˆ’y₁)/(xβ‚‚βˆ’x₁).  |  FOIL = First Β· Outer Β· Inner Β· Last.
Algebra Progress0 / 10
AΒ·01
Linear Equations Β· One Variable
Sarah has 3 times as many stickers as Jake. Together they have 48 stickers. How many stickers does Sarah have?
πŸ’‘ Hint
Let Jake = x. Then Sarah = 3x. Write: x + 3x = 48.
πŸ“– Explanation
Let Jake = x. Then Sarah = 3x.
x + 3x = 48 β†’ 4x = 48 β†’ x = 12 (Jake).
Sarah = 3 Γ— 12 = 36. βœ“ Always define your variable first!
AΒ·02
Two-Step Equations
A taxi charges a $3 flat fee plus $1.50 per mile. If a ride cost $12, how many miles was the ride?
πŸ’‘ Hint
Equation: 3 + 1.5m = 12. Subtract 3 first, then divide by 1.5.
πŸ“– Explanation
3 + 1.5m = 12
1.5m = 9
m = 6 miles.
Key: Subtract the flat fee first, then divide. Don't divide everything by 1.5 right away!
AΒ·03
Slope of a Line
A hiker starts at an elevation of 200 ft and reaches 500 ft after walking 6 miles. What is the average rate of elevation gain (slope) per mile?
πŸ’‘ Hint
Slope = change in height Γ· change in distance = (500 βˆ’ 200) Γ· 6.
πŸ“– Explanation
Slope = Ξ”y / Ξ”x = (500 βˆ’ 200) / (6 βˆ’ 0) = 300 / 6 = 50 ft/mile.
Trap: Don't add 200 + 500. You need the difference (change), not the sum.
AΒ·04
Inequalities
A student needs at least 70 points to pass. She already earned 43 points. What is the minimum score she needs on the next test?
πŸ’‘ Hint
43 + x β‰₯ 70. Solve for x.
πŸ“– Explanation
43 + x β‰₯ 70
x β‰₯ 27
Minimum score = 27.
Key word: "at least" β†’ use β‰₯ (greater than or equal).
AΒ·05
Systems of Equations
Two friends buy snacks. Alex buys 2 chips + 1 drink = $5. Maya buys 1 chip + 2 drinks = $4. How much does ONE drink cost?
πŸ’‘ Hint
Write: 2c + d = 5 and c + 2d = 4. Try elimination or substitution.
πŸ“– Explanation
2c + d = 5 … (1)
c + 2d = 4 … (2)
Multiply (2) by 2: 2c + 4d = 8
Subtract (1): 3d = 3 β†’ d = $1.00
Trick: Multiply one equation to make coefficients match, then subtract.
AΒ·06
Distributive Property
Simplify: \(3(2x - 4) + 5x\)
πŸ’‘ Hint
Distribute 3 to both 2x and βˆ’4 first. Then combine like terms.
πŸ“– Explanation
3(2x βˆ’ 4) + 5x
= 6x βˆ’ 12 + 5x
= 11x βˆ’ 12
Common mistake: Writing 3(2x βˆ’ 4) = 6x βˆ’ 4. Remember to distribute to EVERY term inside!
AΒ·07
Quadratic Equations
A ball is launched upward. Its height in feet after \(t\) seconds is \(h = -16t^2 + 32t\). At what time does the ball hit the ground (h = 0, t > 0)?
πŸ’‘ Hint
Set h = 0: βˆ’16tΒ² + 32t = 0. Factor out βˆ’16t.
πŸ“– Explanation
βˆ’16tΒ² + 32t = 0
βˆ’16t(t βˆ’ 2) = 0
t = 0 (launch) or t = 2 seconds (landing).
Key: Factor out the GCF first. t = 0 is the start, not the answer we want.
AΒ·08
Proportions & Percentages
A store marks up a jacket from $40 to $56. What is the percent increase?
πŸ’‘ Hint
% increase = (new βˆ’ old) Γ· old Γ— 100. Use the ORIGINAL price as the base.
πŸ“– Explanation
% increase = (56 βˆ’ 40) / 40 Γ— 100 = 16/40 Γ— 100 = 40%
Trap: Some students divide by 56 (new price). Always divide by the ORIGINAL value!
AΒ·09
Functions & Function Notation
Given \(f(x) = 2x^2 - 3x + 1\), what is \(f(-1)\)?
πŸ’‘ Hint
Replace every x with (βˆ’1). Be careful with signs: (βˆ’1)Β² = +1.
πŸ“– Explanation
f(βˆ’1) = 2(βˆ’1)Β² βˆ’ 3(βˆ’1) + 1
= 2(1) + 3 + 1
= 2 + 3 + 1 = 6
Trap: βˆ’3(βˆ’1) = +3, not βˆ’3. Negative Γ— negative = positive!
AΒ·10
Factoring Trinomials
Factor completely: \(x^2 - 5x + 6\)
πŸ’‘ Hint
Find two numbers that MULTIPLY to +6 and ADD to βˆ’5.
πŸ“– Explanation
Need: two numbers that multiply to 6 and add to βˆ’5.
βˆ’2 Γ— βˆ’3 = 6 βœ“ and βˆ’2 + (βˆ’3) = βˆ’5 βœ“
Answer: (x βˆ’ 2)(x βˆ’ 3)
Memory: SAME sign β†’ both match middle sign. Different sign β†’ larger number keeps middle sign.
Part 2

Geometry

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QUICK MEMORY POINTS PYTHAGOREAN: aΒ² + bΒ² = cΒ² (c = hypotenuse, always longest).  |  TRIANGLE SUM: angles add to 180Β°.  |  AREA CIRCLE: Ο€rΒ².  |  SIMILAR: same shape, proportional sides.  |  PARALLEL LINES: alternate interior angles are EQUAL.
Geometry Progress0 / 10
GΒ·01
Pythagorean Theorem
A ladder leans against a wall. The base is 6 ft from the wall and the wall is 8 ft tall. How long is the ladder?
πŸ’‘ Hint
Use aΒ² + bΒ² = cΒ². The ladder is c (hypotenuse).
πŸ“– Explanation
6Β² + 8Β² = cΒ²
36 + 64 = 100
c = √100 = 10 ft
Shortcut: Memorize Pythagorean triples: (3,4,5), (6,8,10), (5,12,13). Multiply the triple by a constant!
GΒ·02
Triangle Angle Sum
A triangle has angles measuring 47Β° and 68Β°. What is the measure of the third angle?
πŸ’‘ Hint
All angles in a triangle sum to 180Β°.
πŸ“– Explanation
180Β° βˆ’ 47Β° βˆ’ 68Β° = 65Β°
Rule: Triangle Angle Sum = 180Β°. Always.
GΒ·03
Area of a Circle
A circular pizza has a diameter of 14 inches. What is its area? (Use \(\pi \approx 3.14\))
πŸ’‘ Hint
Area = Ο€rΒ². Diameter = 14, so radius = 7. Don't use 14 as the radius!
πŸ“– Explanation
r = 14 Γ· 2 = 7
A = Ο€ Γ— 7Β² = 3.14 Γ— 49 = 153.86 inΒ²
#1 Mistake: Using diameter (14) instead of radius (7). Diameter Γ· 2 = radius FIRST!
GΒ·04
Similar Triangles
A tree casts a shadow 15 ft long. At the same time, a 5 ft person casts a 3 ft shadow. How tall is the tree?
πŸ’‘ Hint
Set up a proportion: tree height / tree shadow = person height / person shadow.
πŸ“– Explanation
h/15 = 5/3
h = 15 Γ— 5/3 = 25 ft
Key: Corresponding parts must be in the SAME position in each ratio.
GΒ·05
Volume of a Rectangular Prism
A fish tank is 30 cm long, 20 cm wide, and 25 cm tall. What is its volume?
πŸ’‘ Hint
Volume = length Γ— width Γ— height (V = l Γ— w Γ— h).
πŸ“– Explanation
V = 30 Γ— 20 Γ— 25 = 15,000 cmΒ³
Units: Volume is always cubic units (cmΒ³, ftΒ³, etc.). Area uses square units.
GΒ·06
Parallel Lines & Transversals
Two parallel lines are cut by a transversal. One angle measures 115Β°. What is the measure of the alternate interior angle?
πŸ’‘ Hint
Alternate interior angles are EQUAL when lines are parallel. (They are on opposite sides of the transversal, between the parallel lines.)
πŸ“– Explanation
Alternate interior angles are congruent (equal) when lines are parallel.
Answer = 115Β°
Memory: Alt. Interior = EQUAL. Co-interior (same side) = SUPPLEMENTARY (add to 180Β°).
GΒ·07
Perimeter of a Composite Shape
An L-shaped room has outer dimensions 10 m Γ— 8 m, with a 4 m Γ— 3 m rectangle cut from one corner. What is the perimeter of the L-shape?
πŸ’‘ Hint
Cutting a corner does NOT change the total horizontal + vertical distances. The perimeter = same as the full rectangle's perimeter!
πŸ“– Explanation
When a rectangular notch is cut from a corner, the horizontal spans still total 10 m and vertical spans still total 8 m.
Perimeter = 2(10 + 8) = 36 m
Key insight: Cutting a corner adds two new sides that exactly replace the two removed sides in total length.
GΒ·08
Special Right Triangles (45-45-90)
A square has a diagonal of 10 cm. What is the length of one side? (Simplify your answer.)
πŸ’‘ Hint
A square's diagonal splits it into two 45-45-90 triangles. In a 45-45-90 triangle, hypotenuse = side Γ— √2.
πŸ“– Explanation
hypotenuse = side Γ— √2
10 = s√2
s = 10/√2 = 10√2/2 = 5√2 cm
Ratio to memorize: 45-45-90 β†’ sides are x : x : x√2
GΒ·09
Exterior Angles of a Triangle
An exterior angle of a triangle measures 130Β°. The two non-adjacent interior angles are 70Β° and xΒ°. Find x.
πŸ’‘ Hint
Exterior Angle Theorem: exterior angle = sum of the two non-adjacent interior angles.
πŸ“– Explanation
Exterior Angle = sum of two non-adjacent interior angles.
130 = 70 + x
x = 60Β°
Shortcut: Don't use 180Β° βˆ’ 130Β° = 50Β° (that's the interior angle at that vertex, not x).
GΒ·10
Arc Length
A circle has a radius of 9 cm. What is the arc length of a central angle of 80Β°? (Use \(\pi \approx 3.14\), round to nearest tenth.)
πŸ’‘ Hint
Arc length = (central angle / 360Β°) Γ— 2Ο€r. The fraction tells you "what portion of the circle."
πŸ“– Explanation
Arc length = (80/360) Γ— 2 Γ— 3.14 Γ— 9
= (2/9) Γ— 56.52
= 12.6 cm
Formula to memorize: Arc = (ΞΈ/360) Γ— 2Ο€r