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Quantitative Reasoning

Coordinate geometry (lines and slope) — ISEE Upper practice questions

37 questions on the paper
59 practice questions in the app
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Practice 59 Coordinate geometry (lines and slope) questions in the app

What this topic is

Coordinate geometry (lines and slope) on the ISEE Upper Quantitative Reasoning section covers points, intercepts, slope, and the comparison of lines in the plane.

A student must find where a segment meets an axis, count lattice points on a line, and judge whether two given lines are parallel, perpendicular, or intersecting. Items usually appear as short word problems or which-of-the-following statements. Common traps include swapping intercepts, reversing rise over run, and treating distance from the origin as a single coordinate instead of using the distance formula.

Choices typically offer points, whole-number counts, or brief claims about the two lines.

Sample questions

Pick an answer to see whether it is right — nothing is saved, and nothing needs an account.

Question 1Harder

Points with positive integer coordinates lie on the line $x+y=9$ . How many of these points are more than 7 units from the origin?

2✗ Not this one4✓ Correct6✗ Not this one8✗ Not this one

Show answer

Answer: B — 4

The possible points are $(1,8)$ through $(8,1)$ . Only $(1,8)$ , $(2,7)$ , $(7,2)$ , and $(8,1)$ have $x^2+y^2>49$ , so there are 4 such points.

Question 2Mid

Line $p$ passes through $(2,1)$ and $(6,9)$ . Line $q$ passes through $(0,4)$ and $(4,2)$ . Which of the following is true?

p

and

q

are parallel✗ Not this one

p

and

q

are perpendicular✓ Correct

p

and

q

have the same

y

-intercept✗ Not this one

p

and

q

have the same

x

-intercept✗ Not this one

Show answer

Answer: B — $p$ and $q$ are perpendicular

The slope of $p$ is $\dfrac{9-1}{6-2}=2$ . The slope of $q$ is $\dfrac{2-4}{4-0}=-\dfrac{1}{2}$ . The product of the slopes is $2\times\left(-\dfrac{1}{2}\right)=-1$ , so the lines are perpendicular. The intercepts of the two lines are not the same.

Question 3Mid

Segment $\overline{JK}$ joins $J(2,3)$ and $K(-4,-1)$ . At which point does $\overline{JK}$ intersect the $x$ -axis?

(0,0)

✗ Not this one

\left(0,\dfrac{5}{3}\right)

✗ Not this one

\left(-\dfrac{5}{2},0\right)

✓ Correct

\left(\dfrac{5}{2},0\right)

✗ Not this one

Show answer

Answer: C — $\left(-\dfrac{5}{2},0\right)$

The slope of $JK$ is $\dfrac{-1-3}{-4-2}=\dfrac{2}{3}$ , so its equation is $y-3=\dfrac{2}{3}(x-2)$ . On the $x$ -axis, $y=0$ , giving $-3=\dfrac{2}{3}(x-2)$ and $x=-\dfrac{5}{2}$ . The intersection is $\left(-\dfrac{5}{2},0\right)$ .

Every question is picked for where you are right now, wrong answers come back until they stick, and it all works offline.

Practice 59 Coordinate geometry (lines and slope) questions in the app

Coordinate geometry (lines and slope) — ISEE Upper practice questions

What this topic is

Sample questions

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Quantitative Reasoning