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✦ Self-Study Guide 2025

Algebra 1
& Geometry

20 essential problems. Tricky questions. Clear memory points. Study smarter.

20 Problems
2 Topics
4 Choices Each
Score 0 / 0
0 of 20 answered
▸ Part I

Algebra 1 — Word Problems

Equations, inequalities, systems, and real-world applications

A · 01
💡 Key: ISOLATE THE VARIABLE — move numbers to one side
✦ Quick Example
If $2x + 3 = 9$, then $2x = 9 - 3 = 6$, so $x = 3$.
Mia thinks of a number. She multiplies it by 2 and adds 5. The result is 17.
What is her number?
A · 02
💡 Key: D = R × T — distance = rate × time
✦ Quick Example
Car travels at 50 mph for $t$ hours and covers 100 miles: $50t = 100 \Rightarrow t = 2$ hours.
A train travels at 60 miles per hour. How many hours will it take to travel 180 miles? easy
A · 03
💡 Key: PERIMETER = 2(l + w) — rectangle formula
✦ Quick Example
Rectangle with width 3, perimeter 14: $2(l+3)=14 \Rightarrow l+3=7 \Rightarrow l=4$.
The perimeter of a rectangle is 36 cm. Its width is 8 cm. What is its length?
A · 04
💡 Key: COLLECT LIKE TERMS — move x to one side!
✦ Quick Example
$5x - 2 = 3x + 6 \Rightarrow 5x - 3x = 6 + 2 \Rightarrow 2x = 8 \Rightarrow x = 4$
Solve: $3x - 4 = 2x + 7$
⚠ Tricky: Don't forget to move $x$ terms to the same side!
A · 05
💡 Key: LET x = YOUNGER — define your variable clearly
✦ Quick Example
Two ages differ by 3; sum is 25. Let $x$ = younger: $x + (x+3) = 25 \Rightarrow x = 11$.
Alex is 5 years older than Jordan. Together, their ages add up to 41. How old is Jordan? medium
A · 06
💡 Key: PERCENT = (part ÷ whole) × 100
✦ Quick Example
10% of 50 = $0.10 \times 50 = 5$.  Always convert % → decimal first!
A store offers a 15% discount on a $\$80$ jacket.
How much is the discount in dollars?
A · 07
💡 Key: FLIP SIGN when ÷ by negative — inequality rule!
✦ Quick Example
$-3x < 9 \Rightarrow x > -3$  (sign flips when dividing by negative!)
Solve the inequality: $2x + 3 > 11$
⚠ Watch out: when do you flip the sign?
A · 08
💡 Key: COIN PROBLEMS → two equations — count + value
✦ Quick Example
5 nickels + 3 dimes = 5(5) + 3(10) = 25 + 30 = 55¢.  Value = (count) × (coin value).
A piggy bank has 20 coins, all nickels and dimes. The total value is $\$1.40$.
How many dimes are there? medium
A · 09
💡 Key: ELIMINATION — add equations to cancel
✦ Quick Example
$x+y=8$ and $x-y=2$: Add them → $2x=10$ → $x=5$, $y=3$.
$$x + y = 10$$ $$x - y = 4$$ What is the value of $x$?
A · 10
💡 Key: WORK RATE = 1/time — combined rate = sum of rates
✦ Quick Example
A does job in 3 hr, B in 6 hr. Together: $\frac{1}{3}+\frac{1}{6}=\frac{1}{2}$ → done in 2 hr.
Pipe A fills a tank in 4 hours. Pipe B fills it in 6 hours.
Working together, how many hours to fill the tank? medium
📐
▸ Part II

Geometry — Core Problems

Angles, area, volume, similarity, and coordinate geometry

G · 01
💡 Key: TRIANGLE ANGLES SUM = 180°
✦ Quick Example
Angles $2x$, $3x$, $x$: $2x+3x+x = 180 \Rightarrow 6x=180 \Rightarrow x=30$.
A triangle has angles in the ratio $3:1:1$.
What is the value of the smallest angle? easy
G · 02
💡 Key: a² + b² = c² — hypotenuse is always c (longest!)
✦ Quick Example
Legs 5 & 12: $c = \sqrt{5^2 + 12^2} = \sqrt{25+144} = \sqrt{169} = 13$.
A right triangle has legs of length 3 and 4.
What is the length of the hypotenuse?
G · 03
💡 Key: A = πr² — use radius, NOT diameter!
✦ Quick Example
Circle diameter = 10 → radius = 5. Area = $\pi(5)^2 = 25\pi$.
A circle has a diameter of 12.
What is its area? (Leave answer in terms of $\pi$) medium
G · 04
💡 Key: SUPPLEMENTARY = 180°   COMPLEMENTARY = 90°
✦ Quick Example
Angle of 40° is supplementary to: $180° - 40° = 140°$.
An angle measures 65°.
What is its supplementary angle?
⚠ Trap: Don't confuse supplementary (180°) with complementary (90°)!
G · 05
💡 Key: SIMILAR TRIANGLES → equal ratios — set up proportion!
✦ Quick Example
Similar triangles, sides 3 & 5 are corresponding. If bigger has side 10:
$\frac{3}{5} = \frac{?}{10} \Rightarrow ? = 6$.
Two similar triangles have corresponding sides of 4 and 6.
If the smaller triangle has another side of length 6, what is the corresponding side of the larger triangle? medium
G · 06
💡 Key: PARALLELOGRAM A = base × height — NOT the slant side!
✦ Quick Example
Parallelogram base = 6, slant side = 7, height = 4. Area = $6 \times 4 = 24$ (not $6 \times 7$!).
A parallelogram has a base of 8 cm and a height of 5 cm.
What is the area?
G · 07
💡 Key: CONE V = (1/3)πr²h — cone = 1/3 of cylinder!
✦ Quick Example
Cone r=2, h=3: $V = \frac{1}{3}\pi(4)(3) = 4\pi$. Always remember the $\frac{1}{3}$!
Find the volume of a cone with radius $r = 3$ and height $h = 4$.
$$V = \frac{1}{3}\pi r^2 h$$
G · 08
💡 Key: EXTERIOR ANGLE = sum of 2 non-adjacent interior angles
✦ Quick Example
Triangle with interior angles 40° and 55°. Exterior angle = $40° + 55° = 95°$.
A triangle has two interior angles of 50° and 70°.
What is the exterior angle adjacent to the third interior angle? medium
G · 09
💡 Key: MIDPOINT = average the coordinates — add and divide by 2
✦ Quick Example
Midpoint of $(1, 3)$ and $(5, 7)$: $M = \left(\frac{1+5}{2}, \frac{3+7}{2}\right) = (3, 5)$.
Find the midpoint of the segment with endpoints
$A(2, 3)$ and $B(8, 7)$.
G · 10
💡 Key: ARC LENGTH = (θ/360) × 2πr — fraction of full circle
✦ Quick Example
Circle r=6, central angle = 60°. Arc = $\frac{60}{360} \times 2\pi(6) = \frac{1}{6} \times 12\pi = 2\pi$.
A circle has radius 8. A central angle of 90° cuts off an arc.
What is the arc length? (Leave in terms of $\pi$) medium