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PSAT Math : Geometry

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Example Questions

Example Question #4 : Distance Formula

Steven draws a line that is 13 units long. If $(-4,1)$ is one endpoint of the line, which of the following might be the other endpoint?

Possible Answers:

$(3,7)$

$(9,14)$

$(5,12)$

$(1,13)$

$(13,13)$

Correct answer:

$(1,13)$

Explanation:

The distance formula is $\sqrt{((x_{2}-x_{1})^{2} + (y_{2}-y_{1})^{2})}$ .

Plug in $(-4,1)$ with each of the answer choices and solve.

Plug in $(1,13)$ :

$d=\sqrt{(1-(-4))^2+ (13-1)^2} = \sqrt{5^2+12^2}=\sqrt{25+144}=\sqrt{169}=13$

This is therefore the correct answer choice.

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Example Question #5 : Distance Formula

What is the distance between the points (3,12) and (7,2) ?

Possible Answers:

$3\sqrt{5}$

$2\sqrt{29}$

$4\sqrt{2}$

Correct answer:

$2\sqrt{29}$

Explanation:

Plug the points into the distance formula and simplify:

distance² = (x₂ – x₁)² + (y₂ – y₁)² = (7 – 3)² + (2 – 12)² = 4² + 10² = 116

distance = √116 = √(4 * 29) = 2√29

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Example Question #3 : Distance Formula

What is the distance between $(3,4)$ and $(8,16)$ ?

Possible Answers:

$20$

$23$

$12$

$13$

$17$

Correct answer:

$13$

Explanation:

The formula for the distance between two points is $d = \sqrt{(x_{2} - x_{1})^2+(y_{2} - y_{1})^2}$ .

Plug in the points:

$d=\sqrt{(8-3)^2+(16-4)^2}$

$d = \sqrt{5^2+12^2} = 13$

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Example Question #1 : Perpendicular Lines

Which set of lines is perpendicular?

Possible Answers:

y = 3x + 5 and y = 5x + 3

the line between the points (7,4) and (4,7), and the line between the points (3,9) and (4,8)

the line between the points (1,3) and (3,5), and y = 4x + 7

y = x – 1/2 and y = –x + 1/2

y = 3x/5 – 3 and y = 5x/3 + 3

Correct answer:

y = x – 1/2 and y = –x + 1/2

Explanation:

Two lines are perpendicular to each other if their slopes are negative reciprocals. For example, if one line has a slope of 2, the line perpendicular to it has a slope of –1/2. One easy way to eliminate answer choices is to check if the slopes have the same sign, i.e. both positive or both negative. If so, they cannot be perpendicular. Several of the lines in the answer choices are of the form y = mx + b, where m is the slope and b is the y-intercept. We are only worried about the slope for the purposes of this question.

y = 3x + 5 and y = 5x + 3 both have positive slopes (m = 3 and m = 5, respectively), so they aren't perpendicular.

y = 3x/5 – 3 and y = 5x/3 + 3 both have positive slopes, so again they aren't perpendicular.

y = x – 1/2 and y = –x + 1/2 have slopes of m = 1 and m = –1, respectively. One is positive and one is negative, so that is a good sign. Let's take the negative reciprocal of 1. 1 /–1 = –1. So these two slopes are in fact negative reciprocals, and these two lines are perpendicular to each other. Even though we have found the correct answer, let's go through the other two choices to be sure.

The line between the points (1,3) and (3,5), and y = 4x + 7: We need to find the slope of the first line. slope = rise / run = (y₂ – y₁) / (x₂ – x₁) = (5 – 3) / (3 – 1) = 1. The slope of y = 4x + 7 is also positive (m = 4), so the lines are not perpendicular.

The line between the points (7,4) and (4,7), and the line between the points (3,9) and (4,8): the first slope = (7 – 4) / (4 – 7) = –1 and the second slope = (8 – 9) / (4 – 3) = –1. They have the same slope, making them parallel, not perpendicular.

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Example Question #41 : Perpendicular Lines

Which set of lines is perpendicular?

Possible Answers:

y = x – 1/2 and y = –x + 1/2

the line between the points (1,3) and (3,5), and y = 4x + 7

the line between the points (7,4) and (4,7), and the line between the points (3,9) and (4,8)

y = 3x + 5 and y = 5x + 3

y = 3x/5 – 3 and y = 5x/3 + 3

Correct answer:

y = x – 1/2 and y = –x + 1/2

Explanation:

y = 3x + 5 and y = 5x + 3 both have positive slopes (m = 3 and m = 5, respectively), so they aren't perpendicular.

y = 3x/5 – 3 and y = 5x/3 + 3 both have positive slopes, so again they aren't perpendicular.

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Example Question #1 : How To Find Out If Lines Are Perpendicular

Which of the following lines is perpendicular to $y=3x-4$

Possible Answers:

$y=-3x-4$

$y=\frac{1}{3}x-4$

$y=\frac{1}{3}x+4$

$y=-\frac{1}{3}x-4$

Correct answer:

$y=-\frac{1}{3}x-4$

Explanation:

The line which is perpendicular has a slope which is the negative inverse of the slope of the original line.

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Example Question #1 : How To Find Out If Lines Are Perpendicular

If two lines have slopes of $-5$ and $\frac{1}{5}$ , which statement about the lines is true?

Possible Answers:

They are parallel.

They intersect at two points.

They are parabolas.

They don't intersect.

They are perpendicular.

Correct answer:

They are perpendicular.

Explanation:

Perpendicular lines have slopes that are the negative reciprocals of each other.

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Example Question #5 : Perpendicular Lines

Line includes the points (4, -3) and (8, -6) . Line includes the points (-12, 9) and (-20, 3) . Which of the following statements is true of these lines?

Possible Answers:

The lines are identical.

More information is needed to answer this question.

The lines are perpendicular.

The lines are parallel.

The lines are distinct but neither parallel nor perpendicular.

Correct answer:

The lines are distinct but neither parallel nor perpendicular.

Explanation:

We calculate the slopes of the lines using the slope formula.

The slope of line is

$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} = \frac{-3 - (-6)}{4-8} = \frac{3}{-4} = - \frac{3}{4}$

The slope of line is

$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} = \frac{3- 9}{-20- (-12)} = \frac{-6}{-8} = \frac{3}{4}$

Parallel lines and identical lines must have the same slope, so these can be eliminated as choices. The slopes of perpendicular lines must have product . The slopes have product

$- \frac{3}{4} \times \frac{3}{4} = - \frac{9}{16} \neq -1$

so they are not perpendicular.

The correct response is that the lines are distinct but neither parallel nor perpendicular.

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Example Question #2 : Perpendicular Lines

Line includes the points (4, -3) and (8, -6) . Line includes the points (5, -2) and (11, 6) . Which of the following statements is true of these lines?

Possible Answers:

The lines are perpendicular.

The lines are distinct but neither parallel nor perpendicular.

The lines are parallel.

Insufficient information is given to answer this question.

The lines are identical.

Correct answer:

The lines are perpendicular.

Explanation:

We calculate the slopes of the lines using the slope formula.

The slope of line is

$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} = \frac{-3 - (-6)}{4-8} = \frac{3}{-4} = - \frac{3}{4}$

The slope of line is

$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} = \frac{-2 - 6}{5- 11} = \frac{-8}{-6} = \frac{4}{3}$

The slopes are not the same, so the lines are neither parallel nor identical. We multiply their slopes to test for perpendicularity:

$- \frac{3}{4} \cdot \frac{4} {3} = - \frac{12}{12} = -1$

The product of the slopes is , making the lines perpendicular.

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Example Question #1 : Perpendicular Lines

Consider the equations y = 4x + 9 and 8x - 2y = 18 . Which of the following statements is true of the lines of these equations?

Possible Answers:

The lines are distinct but neither parallel nor perpendicular.

The lines are identical.

The lines are perpendicular.

The lines are parallel.

Insufficient information is given to answer this question.

Correct answer:

The lines are parallel.

Explanation:

We find the slope of each line by putting each equation in slope-intercept form y =mx + b and examining the coefficient of .

y = 4x + 9 is already in slope-intercept form; its slope is m = 4 .

To get 8x - 2y = 18 in slope-intercept form we solve for :

8x - 2y = 18

8x - 2y - 8x = 18 - 8x

- 2y = - 8x + 18

$\frac{- 2y }{-2}= \frac{- 8x + 18}{-2}$

y =4x -9

The slope of this line is also m = 4 .

The slopes are equal; however, the -intercepts are different - the -intercept of the first line is (0,9) and that of the second line is (0,-9) . Therefore, the lines are parallel as opposed to being the same line.

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PSAT Math : Geometry

Study concepts, example questions & explanations for PSAT Math

All PSAT Math Resources

Example Questions

Example Question #4 : Distance Formula

Example Question #5 : Distance Formula

Example Question #3 : Distance Formula

Example Question #1 : Perpendicular Lines

Example Question #41 : Perpendicular Lines

Example Question #1 : How To Find Out If Lines Are Perpendicular

Example Question #1 : How To Find Out If Lines Are Perpendicular

Example Question #5 : Perpendicular Lines

Example Question #2 : Perpendicular Lines

Example Question #1 : Perpendicular Lines

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