ALGEBRA 1 — Linear Functions
10 Questions
slope = rise ÷ run = (y₂−y₁) ÷ (x₂−x₁) |
slope-intercept y = mx + b |
vertical line x = a (UNDEFINED slope) |
horizontal line y = b (slope = 0) |
parallel same slope |
perpendicular slopes are negative reciprocals
ALGEBRA · Q1 ⭐ Easy
What is the slope of the line that passes through \((2, 5)\) and \((6, 13)\)?
💭 Hint: Use the slope formula → m = (y₂ − y₁) / (x₂ − x₁)
slope \(m = \dfrac{13-5}{6-2} = \dfrac{8}{4} = 2\)
✅ Answer: B
Count rise (↑8) and run (→4) → 8÷4 = 2
✅ Answer: B
Count rise (↑8) and run (→4) → 8÷4 = 2
ALGEBRA · Q2 ⭐ Easy
Which equation represents a line with slope \(-3\) and y-intercept \(7\)?
💭 Hint: y = mx + b format → m is slope, b is y-intercept
\(y = \underbrace{-3}_{slope}x + \underbrace{7}_{y\text{-int}}\)
✅ Answer: A
In \(y=mx+b\): \(m\) goes with \(x\), \(b\) stands alone!
✅ Answer: A
In \(y=mx+b\): \(m\) goes with \(x\), \(b\) stands alone!
ALGEBRA · Q3 ⭐⭐ Medium
A vertical line passes through the point \((4, -2)\). What is its equation?
💭 KEY: Vertical line → x = constant (slope is UNDEFINED!)
Vertical line: ALL points have the same x-value → \(x = 4\)
✅ Answer: A
🚫 Trick: Don't confuse with \(y=-2\) (that's a HORIZONTAL line!)
✅ Answer: A
🚫 Trick: Don't confuse with \(y=-2\) (that's a HORIZONTAL line!)
ALGEBRA · Q4 ⭐⭐ Medium
What is the slope of the line \(3x - 6y = 12\)?
💭 Rewrite in y = mx + b form first! Isolate y.
\(3x - 6y = 12\)
\(-6y = -3x + 12\)
\(y = \dfrac{1}{2}x - 2\) → slope \(= \dfrac{1}{2}\)
✅ Answer: C
⚠️ Trap: Don't just read the "3" — always convert to \(y=mx+b\) first!
\(-6y = -3x + 12\)
\(y = \dfrac{1}{2}x - 2\) → slope \(= \dfrac{1}{2}\)
✅ Answer: C
⚠️ Trap: Don't just read the "3" — always convert to \(y=mx+b\) first!
ALGEBRA · Q5 ⭐⭐ Medium
Line \(L\) has equation \(y = 4x + 1\). Which line is parallel to \(L\)?
💭 PARALLEL = SAME slope! Different y-intercepts.
Parallel → same slope \(m=4\), but different b
\(y = 4x - 5\) has slope 4 ✓ and different intercept ✓
✅ Answer: C
⚠️ D is the same line (identical), not just parallel!
\(y = 4x - 5\) has slope 4 ✓ and different intercept ✓
✅ Answer: C
⚠️ D is the same line (identical), not just parallel!
ALGEBRA · Q6 ⭐⭐⭐ Tricky!
Line \(A\) has slope \(\dfrac{2}{3}\). What is the slope of a line perpendicular to \(A\)?
💭 PERPENDICULAR: Flip the fraction AND change sign → negative reciprocal
Perpendicular slope = negative reciprocal of \(\dfrac{2}{3}\)
Step 1: Flip → \(\dfrac{3}{2}\)
Step 2: Change sign → \(-\dfrac{3}{2}\)
Check: \(\dfrac{2}{3} \times (-\dfrac{3}{2}) = -1\) ✓
✅ Answer: D
💡 Always multiply: if result = −1, they're perpendicular!
Step 1: Flip → \(\dfrac{3}{2}\)
Step 2: Change sign → \(-\dfrac{3}{2}\)
Check: \(\dfrac{2}{3} \times (-\dfrac{3}{2}) = -1\) ✓
✅ Answer: D
💡 Always multiply: if result = −1, they're perpendicular!
ALGEBRA · Q7 ⭐⭐ Medium
What is the x-intercept of the line \(y = 2x - 8\)?
💭 x-intercept → set y = 0 and solve for x
Set \(y = 0\): \(0 = 2x - 8 \Rightarrow 2x = 8 \Rightarrow x = 4\)
x-intercept = \((4, 0)\)
✅ Answer: B
⚠️ Trap: D is the y-intercept (\(x=0\) → \(y=-8\)) — don't mix them up!
x-intercept = \((4, 0)\)
✅ Answer: B
⚠️ Trap: D is the y-intercept (\(x=0\) → \(y=-8\)) — don't mix them up!
ALGEBRA · Q8 ⭐⭐⭐ Tricky!
Which of these lines has an undefined slope?
💭 UNDEFINED slope = vertical line = equation is x = number
• \(y=5\) → horizontal, slope = 0
• \(x=-3\) → vertical → slope = UNDEFINED ✓
• \(y=0\) → x-axis, slope = 0
• \(y=x\) → slope = 1
✅ Answer: B
💡 Memory: "x = number" lines are vertical = undefined!
• \(x=-3\) → vertical → slope = UNDEFINED ✓
• \(y=0\) → x-axis, slope = 0
• \(y=x\) → slope = 1
✅ Answer: B
💡 Memory: "x = number" lines are vertical = undefined!
ALGEBRA · Q9 ⭐⭐⭐ Tricky!
Write the equation of the line through \((0, 3)\) and \((4, 0)\) in slope-intercept form.
💭 y-intercept is given directly! Then find slope from the two points.
y-intercept: point \((0,3)\) → \(b = 3\)
slope: \(m = \dfrac{0-3}{4-0} = \dfrac{-3}{4}\)
Equation: \(y = -\dfrac{3}{4}x + 3\)
✅ Answer: B
⚠️ Trap: Going from (0,3) to (4,0) is going DOWN-right → negative slope!
slope: \(m = \dfrac{0-3}{4-0} = \dfrac{-3}{4}\)
Equation: \(y = -\dfrac{3}{4}x + 3\)
✅ Answer: B
⚠️ Trap: Going from (0,3) to (4,0) is going DOWN-right → negative slope!
ALGEBRA · Q10 ⭐⭐⭐ Challenge!
Line \(P\) passes through \((-1, 4)\) and is perpendicular to \(y = \dfrac{1}{2}x - 3\). What is the equation of line \(P\)?
💭 Step 1: Find perp. slope. Step 2: Use point-slope form y − y₁ = m(x − x₁)
Original slope: \(\frac{1}{2}\) → Perp. slope: \(-2\)
Use \((-1, 4)\): \(y - 4 = -2(x - (-1))\)
\(y - 4 = -2x - 2\)
\(y = -2x + 2\)
✅ Answer: A
💡 Check: \(\frac{1}{2} \times (-2) = -1\) ✓ perpendicular confirmed!
Use \((-1, 4)\): \(y - 4 = -2(x - (-1))\)
\(y - 4 = -2x - 2\)
\(y = -2x + 2\)
✅ Answer: A
💡 Check: \(\frac{1}{2} \times (-2) = -1\) ✓ perpendicular confirmed!
— End of Algebra 1 Section · 10 Questions —
GEOMETRY — Shapes, Angles & Theorems
10 Questions
Triangle ∑ angles = 180° |
Quadrilateral = 360° |
Pythagorean a²+b²=c² |
Area △ = ½bh |
Area ⬜ = lw |
Supplementary = 180° |
Complementary = 90° |
Vertical angles = equal
GEOMETRY · Q1 ⭐ Easy
Two angles are supplementary. One angle measures \(65°\). What is the other angle?
💭 Supplementary = two angles ADD UP to 180°
Supplementary: \(65° + x = 180°\)
\(x = 180° - 65° = 115°\)
✅ Answer: C
⚠️ Trap: 25° is the complement (adds to 90°), not supplement!
\(x = 180° - 65° = 115°\)
✅ Answer: C
⚠️ Trap: 25° is the complement (adds to 90°), not supplement!
GEOMETRY · Q2 ⭐ Easy
A triangle has angles \(47°\) and \(83°\). What is the third angle?
💭 Triangle rule: All 3 angles ALWAYS add up to 180°
\(47° + 83° + x = 180°\)
\(130° + x = 180°\)
\(x = 50°\)
✅ Answer: B
\(130° + x = 180°\)
\(x = 50°\)
✅ Answer: B
GEOMETRY · Q3 ⭐⭐ Medium
A right triangle has legs of length \(6\) and \(8\). What is the hypotenuse?
💭 Pythagorean Theorem: a² + b² = c² (c = hypotenuse, always opposite the right angle)
\(a^2 + b^2 = c^2\)
\(6^2 + 8^2 = c^2\)
\(36 + 64 = 100\)
\(c = \sqrt{100} = 10\)
✅ Answer: A
💡 3-4-5 and 6-8-10 are classic Pythagorean triples — memorize them!
\(6^2 + 8^2 = c^2\)
\(36 + 64 = 100\)
\(c = \sqrt{100} = 10\)
✅ Answer: A
💡 3-4-5 and 6-8-10 are classic Pythagorean triples — memorize them!
GEOMETRY · Q4 ⭐⭐ Medium
Two lines intersect forming vertical angles. One angle is \(42°\). What are the other three angles?
💭 Vertical angles are EQUAL. Adjacent angles are supplementary (add to 180°).
Vertical angles = equal: opposite angle = 42°
Adjacent (supplementary): \(180° - 42° = 138°\)
So: 42°, 138°, 42°, 138°
✅ Answer: A (the other three: 42°, 138°, 138°)
💡 X shape → opposite = same, adjacent = 180°
Adjacent (supplementary): \(180° - 42° = 138°\)
So: 42°, 138°, 42°, 138°
✅ Answer: A (the other three: 42°, 138°, 138°)
💡 X shape → opposite = same, adjacent = 180°
GEOMETRY · Q5 ⭐⭐ Medium
What is the area of a triangle with base \(12\) cm and height \(9\) cm?
💭 Area of triangle = ½ × base × height
\(A = \dfrac{1}{2} \times 12 \times 9 = \dfrac{1}{2} \times 108 = 54 \text{ cm}^2\)
✅ Answer: B
⚠️ Trap: A (108) is the area WITHOUT the ½ — always divide by 2!
✅ Answer: B
⚠️ Trap: A (108) is the area WITHOUT the ½ — always divide by 2!
GEOMETRY · Q6 ⭐⭐⭐ Tricky!
In the figure, a transversal crosses two parallel lines. If one angle is \(110°\), what is its alternate interior angle?
💭 Alternate interior angles: between the lines, on OPPOSITE sides → they are EQUAL
Alternate interior angles (parallel lines) = EQUAL
So the alternate interior angle = \(110°\)
✅ Answer: B
💡 Memory: COrresponding = Equal, Alternate = Equal, Co-interior = 180°
So the alternate interior angle = \(110°\)
✅ Answer: B
💡 Memory: COrresponding = Equal, Alternate = Equal, Co-interior = 180°
GEOMETRY · Q7 ⭐⭐ Medium
What is the perimeter of a rectangle with length \(15\) m and width \(7\) m?
💭 Perimeter = 2(length + width) OR add all 4 sides
\(P = 2(15 + 7) = 2 \times 22 = 44 \text{ m}\)
✅ Answer: B
⚠️ Trap: A (105) is the AREA \(15 \times 7\). Know the difference: perimeter=border, area=inside!
✅ Answer: B
⚠️ Trap: A (105) is the AREA \(15 \times 7\). Know the difference: perimeter=border, area=inside!
GEOMETRY · Q8 ⭐⭐⭐ Tricky!
A polygon has interior angles summing to \(720°\). How many sides does it have?
💭 Formula: Sum of interior angles = (n − 2) × 180°, where n = number of sides
\((n-2) \times 180 = 720\)
\(n - 2 = \dfrac{720}{180} = 4\)
\(n = 6\) → Hexagon!
✅ Answer: B
💡 Quick check: triangle=180°, quad=360°, pent=540°, hex=720°
\(n - 2 = \dfrac{720}{180} = 4\)
\(n = 6\) → Hexagon!
✅ Answer: B
💡 Quick check: triangle=180°, quad=360°, pent=540°, hex=720°
GEOMETRY · Q9 ⭐⭐⭐ Tricky!
Triangle \(ABC\) has an exterior angle at vertex \(C\) measuring \(120°\). The interior angles at \(A\) and \(B\) are equal. What is the measure of angle \(A\)?
💭 Exterior angle = sum of the TWO non-adjacent interior angles (Exterior Angle Theorem)
Exterior Angle Theorem: ext. angle = sum of remote interior angles
\(\angle A + \angle B = 120°\)
Since \(\angle A = \angle B\): \(2\angle A = 120° \Rightarrow \angle A = 60°\)
✅ Answer: C
💡 Always: exterior angle = sum of the OTHER two interior angles!
\(\angle A + \angle B = 120°\)
Since \(\angle A = \angle B\): \(2\angle A = 120° \Rightarrow \angle A = 60°\)
✅ Answer: C
💡 Always: exterior angle = sum of the OTHER two interior angles!
GEOMETRY · Q10 ⭐⭐⭐ Challenge!
In a right triangle, one leg is \(5\) and the hypotenuse is \(13\). What is the area of the triangle?
💭 Step 1: Find the other leg using Pythagorean theorem. Step 2: Area = ½ × leg₁ × leg₂
Step 1: Find other leg:
\(5^2 + b^2 = 13^2 \Rightarrow 25 + b^2 = 169 \Rightarrow b^2 = 144 \Rightarrow b = 12\)
Step 2: Area = \(\dfrac{1}{2} \times 5 \times 12 = 30\)
✅ Answer: A
💡 5-12-13 is a classic Pythagorean triple! Memorize it!
\(5^2 + b^2 = 13^2 \Rightarrow 25 + b^2 = 169 \Rightarrow b^2 = 144 \Rightarrow b = 12\)
Step 2: Area = \(\dfrac{1}{2} \times 5 \times 12 = 30\)
✅ Answer: A
💡 5-12-13 is a classic Pythagorean triple! Memorize it!
— End of Geometry Section · 10 Questions —