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

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

What this topic is

Coordinate geometry (lines and slope) asks about points and lines on the xy-plane.

Parallel lines have equal slopes, and perpendicular lines have slopes that are negative reciprocals, so a line with slope two is perpendicular to one with slope negative one-half. Some questions ask where a segment crosses an axis, which means finding the line's equation and setting y equal to zero for the x-axis.

Others combine a line with distance from the origin, using the Pythagorean theorem. Choices often pair statements such as the lines are parallel or the lines are perpendicular, so computing both slopes exactly is the reliable path.

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

Question 3Mid

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

(0, 0)

✗ Not this one

(0, \frac{5}{3})

✗ Not this one

(- \frac{5}{2}, 0)

✓ Correct

(\frac{5}{2}, 0)

✗ Not this one

Show answer

Answer: C — $(- \frac{5}{2}, 0)$

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

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