Coordinate geometry (midpoint, distance) — ISEE Upper practice questions

Coordinate geometry (midpoint, distance) on the ISEE Upper Mathematics Achievement section covers midpoints, distances between points, axis reflections, and perpendicular lines in the plane. A student must find a midpoint from two endpoints, compute distance, including vertical distance, reflect a point across an axis, and identify which point lies on a line perpendicular to a given line through a given point. Items usually ask for a point, a distance, or a reflected pair, with other points or lengths as choices; traps include swapping coordinates, reflecting across the wrong axis, and using a parallel slope instead of the negative reciprocal.

Section: Mathematics Achievement 47 questions on the paper
88 practice questions in the app

Sample questions

Question 1

Line $k$ passes through $(2,-1)$ and $(8,3)$ . Which point lies on a line perpendicular to $k$ that passes through $(2,-1)$ ?

A $(6,5)$
B $(-4,7)$
C $(-4,-5)$
D $(6,-7)$

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Answer: D — $(6,-7)$

The slope of $k$ is $\dfrac{3-(-1)}{8-2}=\dfrac46=\dfrac23$ . A perpendicular line has slope $-\dfrac32$ . From $(2,-1)$ to $(6,-7)$ , the slope is $\dfrac{-7-(-1)}{6-2}=\dfrac{-6}{4}=-\dfrac32$ , so $(6,-7)$ lies on the perpendicular line.

Question 2

What is the reflection of $(2,6)$ across the $x$ -axis?

A $(-2,6)$
B $(2,-6)$
C $(-2,-6)$
D $(6,2)$

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Answer: B — $(2,-6)$

Reflection over the $x$ -axis keeps x and negates y, producing $(2,-6)$ .

Question 3

What is the vertical distance between $R(-2,2)$ and $S(-2,8)$ ?

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Answer: B — 6

The x-coordinates match, so distance is $|8-(2)|=6$ .

Practice 88 Coordinate geometry (midpoint, distance) questions in the app

Coordinate geometry (midpoint, distance) — ISEE Upper practice questions

Sample questions

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Mathematics Achievement