📐 Geometry Coordinate Geometry
Recommended: 30–40 minutes. No calculator.
Problems
1.
Tags: Distance Formula · Easy · source: Original (AMC-style)
What is the distance between (0,0) and (3,4)?
A) $3$ B) $4$ C) $5$ D) $6$ E) $7$
Answer & Solution
Answer: C
Distance = $\sqrt{(3-0)^2 + (4-0)^2} = \sqrt{9 + 16} = 5$.
2.
Tags: Midpoint Formula · Easy · source: Original (AMC-style)
What is the midpoint of (1,3) and (5,7)?
A) $(2,4)$ B) $(3,5)$ C) $(4,6)$ D) $(5,7)$ E) $(6,8)$
Answer & Solution
Answer: B
Midpoint = $\left(\frac{1+5}{2}, \frac{3+7}{2}\right) = (3,5)$.
3.
Tags: Slope · Easy · source: Original (AMC-style)
What is the slope of the line through (1,2) and (4,8)?
A) $1$ B) $2$ C) $3$ D) $4$ E) $6$
Answer & Solution
Answer: B
Slope = $\frac{8-2}{4-1} = \frac{6}{3} = 2$.
4.
Tags: Equation of Line · Easy · source: Original (AMC-style)
What is the equation of the line through (0,3) with slope 2?
A) $y = 2x$ B) $y = 2x + 3$ C) $y = 3x + 2$ D) $y = x + 3$ E) $y = 2x - 3$
Answer & Solution
Answer: B
Using point-slope form: $y - 3 = 2(x - 0)$, so $y = 2x + 3$.
5.
Tags: Parallel Lines · Easy · source: Original (AMC-style)
What is the slope of a line parallel to $y = 3x + 1$?
A) $-3$ B) $-1$ C) $1$ D) $3$ E) $\frac{1}{3}$
Answer & Solution
Answer: D
Parallel lines have the same slope, so the slope is $3$.
6.
Tags: Perpendicular Lines · Easy · source: Original (AMC-style)
What is the slope of a line perpendicular to $y = 2x + 1$?
A) $-2$ B) $-\frac{1}{2}$ C) $\frac{1}{2}$ D) $2$ E) $4$
Answer & Solution
Answer: B
Perpendicular lines have slopes that are negative reciprocals, so the slope is $-\frac{1}{2}$.
7.
Tags: Distance to Origin · Medium · source: Original (AMC-style)
What is the distance from (3,4) to the origin?
A) $3$ B) $4$ C) $5$ D) $6$ E) $7$
Answer & Solution
Answer: C
Distance = $\sqrt{3^2 + 4^2} = \sqrt{9 + 16} = 5$.
8.
Tags: Circle Equation · Medium · source: Original (AMC-style)
What is the equation of a circle with center (2,3) and radius 4?
A) $(x-2)^2 + (y-3)^2 = 4$ B) $(x+2)^2 + (y+3)^2 = 4$ C) $(x-2)^2 + (y-3)^2 = 16$ D) $(x+2)^2 + (y+3)^2 = 16$ E) $x^2 + y^2 = 16$
Answer & Solution
Answer: C
The equation is $(x-h)^2 + (y-k)^2 = r^2$, so $(x-2)^2 + (y-3)^2 = 16$.
9.
Tags: Area of Triangle · Medium · source: Original (AMC-style)
What is the area of triangle with vertices (0,0), (3,0), and (0,4)?
A) $3$ B) $4$ C) $6$ D) $8$ E) $12$
Answer & Solution
Answer: C
Using the formula: area = $\frac{1}{2}|x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2)| = \frac{1}{2}|0(0-4) + 3(4-0) + 0(0-0)| = \frac{1}{2} \times 12 = 6$.
10.
Tags: Intersection of Lines · Medium · source: Original (AMC-style)
What is the intersection point of $y = 2x + 1$ and $y = -x + 4$?
A) $(0,1)$ B) $(1,3)$ C) $(2,5)$ D) $(3,7)$ E) $(4,9)$
Answer & Solution
Answer: B
Setting equal: $2x + 1 = -x + 4$, so $3x = 3$ and $x = 1$. Then $y = 2(1) + 1 = 3$, so the point is $(1,3)$.
11.
Tags: Reflection · Hard · source: Original (AMC-style)
What is the reflection of (3,4) across the x-axis?
A) $(3,-4)$ B) $(-3,4)$ C) $(-3,-4)$ D) $(4,3)$ E) $(-4,-3)$
Answer & Solution
Answer: A
Reflecting across the x-axis changes the sign of the y-coordinate, so $(3,4) \to (3,-4)$.
12.
Tags: Rotation · Hard · source: Original (AMC-style)
What is the result of rotating (1,0) 90° counterclockwise about the origin?
A) $(0,1)$ B) $(0,-1)$ C) $(-1,0)$ D) $(1,1)$ E) $(-1,1)$
Answer & Solution
Answer: A
A 90° counterclockwise rotation maps $(x,y) \to (-y,x)$, so $(1,0) \to (0,1)$.
Answer Key
| # | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Ans | C | B | B | B | D | B | C | C | C | B | A | A |