Coordinate Geometry Practice Questions

Question 1 of 5 Medium

In the figure shown, point $P(-\sqrt{3}, 1)$ and $Q(s, t)$ lie on the circle with center O. What is the value of $s$?

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Correct answer: B

As shown in the figure, OP and OQ are perpendicular radii of the circle with center O at the origin.

The slope of line OP is $-\frac{1}{\sqrt{3}}$.

Since OP ⊥ OQ, their slopes are negative reciprocals. The slope of OQ is $\frac{t}{s}$, so: 
\[\frac{t}{s} \times \left(-\frac{1}{\sqrt{3}}\right) = -1\] 
\[t = \sqrt{3}s\]

Since OP is a radius: $OP = \sqrt{(-\sqrt{3})^2 + 1^2} = \sqrt{3 + 1} = 2$

Since OQ = OP = 2: $t^2 + s^2 = 4$

Substituting $t = \sqrt{3}s$: 
\[(\sqrt{3}s)^2 + s^2 = 4\] 
\[3s^2 + s^2 = 4\] 
\[4s^2 = 4\] 
\[s^2 = 1\] 
\[s = \pm 1\]

Since point Q is shown in the first quadrant in the figure (where both coordinates are positive), $s = 1$.

The answer is B.

Question 2 of 5 Medium

For any triangle $ T $ in the xy-coordinate plane, the center of $ T $ is defined to be the point whose x-coordinate is the average (arithmetic mean) of the x-coordinates of the vertices of $ T $ and whose y-coordinate is the average of the y-coordinates of the vertices of $ T $. If a certain triangle has vertices at the points $ (0,0) $ and $ (6,0) $ and center at the point $ (3,2) $, what are the coordinates of the remaining vertex?

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Correct answer: B

Let the third vertex be $ (x, y) $. Center x: $$ \frac{0 + 6 + x}{3} = 3 \implies 6 + x = 9 \implies x = 3 $$ Center y: $$ \frac{0 + 0 + y}{3} = 2 \implies y = 6 $$ Vertex is $ (3, 6) $. The correct answer is B.

Question 3 of 5 Medium

In the xy-plane, line $ k $ has positive slope and x-intercept 4. If the area of the triangle formed by line $ k $ and the two axes is 12, what is the y-intercept of line $ k $?

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Correct answer: A

Triangle vertices are $ (0,0) $, $ (4,0) $, and $ (0,y) $. Area: $$ \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 4 \times |y| = 2|y| $$ $$ 2|y| = 12 \implies |y| = 6 $$ Slope: $$ \frac{y-0}{0-4} = -\frac{y}{4} $$ Since slope is positive, $ -\frac{y}{4} > 0 \implies y < 0 $. So $ y = -6 $. The correct answer is A.

Question 4 of 5 Medium

Line $ l $ is defined by the equation $ y - 5x = 4 $ and line $ w $ is defined by the equation $ 10y + 2x + 20 = 0 $. If line $ k $ does not intersect line $ l $, what is the degree measure of the angle formed by line $ k $ and line $ w $?

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Question 5 of 5 Medium

In the rectangular coordinate plane, points $ X $ and $ Z $ lie on the same line through the origin and points $ W $ and $ Y $ lie on the same line through the origin. If $ a^2 + b^2 = c^2 + d^2 $ and $ e^2 + f^2 = g^2 + h^2 $, what is the value of length $ XZ - WY $?

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