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

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.

Section: Quantitative Reasoning 37 questions on the paper
59 practice questions in the app

Sample questions

Question 1

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?

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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 2

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

A $p$ and $q$ are parallel
B $p$ and $q$ are perpendicular
C $p$ and $q$ have the same $y$ -intercept
D $p$ and $q$ have the same $x$ -intercept

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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 3

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

A $(0,0)$
B $\left(0,\dfrac{5}{3}\right)$
C $\left(-\dfrac{5}{2},0\right)$
D $\left(\dfrac{5}{2},0\right)$

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

Parametrize from $C$ by $(2,3)+t(-6,-4)$ . Setting the $y$ -coordinate equal to 0 gives $3-4t=0$ , so $t=\dfrac{3}{4}$ . The $x$ -coordinate is then $2+\dfrac{3}{4}(-6)=-\dfrac{5}{2}$ .

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

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

Sample questions

More ISEE Upper topics

Quantitative Reasoning