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PSAT Math : Geometry

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Example Questions

Example Question #1 : How To Find The Equation Of A Perpendicular Line

What is the equation for the line that is perpendicular to 4x - 3y = 6 through point (4, 6) ?

Possible Answers:

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

$y = \frac{-3}{4}x +2$

$y = \frac{-3}{4}x +9$

$y = \frac{4}{3}x + 6$

$y = \frac{4}{3}x - 5$

Correct answer:

$y = \frac{-3}{4}x +9$

Explanation:

Perpendicular slopes are opposite reciprocals.

The given slope is found by converting the equation to the slope-intercept form.

4x - 3y = 6

$y = \frac{4}{3}x - 2$

The slope of the given line is $m = \frac{4}{3}$ and the perpendicular slope is $m = \frac{-3}{4}$ .

We can use the given point and the new slope to find the perpendicular equation. Plug in the slope and the given coordinates to solve for the y-intercept.

$6 = \frac{-3}{4}(4) + b$

b = 9

Using this y-intercept in slope-intercept form, we get out final equation: $y = \frac{-3}{4}x + 9$ .

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Example Question #2 : How To Find The Equation Of A Perpendicular Line

Which line below is perpendicular to 5x+6y=18 ?

Possible Answers:

$y=-\frac{5}{6}x+\frac{6}{5}$

$y=\frac{5}{6}x+2$

$y = \frac{6}{5}x + 3$

$y=\frac{5}{6}x+\frac{6}{5}$

$y=-\frac{6}{5}x+8$

Correct answer:

$y = \frac{6}{5}x + 3$

Explanation:

The definition of a perpendicular line is one that has a negative, reciprocal slope to another.

For this particular problem, we must first manipulate our initial equation into a more easily recognizable and useful form: slope-intercept form or y=mx+b .

5x+6y=18

6y=-5x+18

$y=-\frac{5}{6}x+6$

According to our y=mx+b formula, our slope for the original line is $-\frac{5}{6}$ . We are looking for an answer that has a perpendicular slope, or an opposite reciprocal. The opposite reciprocal of $-\frac{5}{6}$ is $\frac{6}{5}$ . Flip the original and multiply it by .

Our answer will have a slope of $\frac{6}{5}$ . Search the answer choices for $\frac{6}{5}$ in the position of the y=mx+b equation.

$\dpi{100} y = \frac{6}{5}x + 3$ is our answer.

(As an aside, the negative reciprocal of 4 is $-\frac{1}{4}$ . Place the whole number over one and then flip/negate. This does not apply to the above problem, but should be understood to tackle certain permutations of this problem type where the original slope is an integer.)

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Example Question #1 : How To Find The Equation Of A Perpendicular Line

If a line has an equation of $2y=3x+3$ , what is the slope of a line that is perpendicular to the line?

Possible Answers:

$-\frac{3}{2}$

$-\frac{2}{3}$

$\frac{3}{2}$

$-2$

$3$

Correct answer:

$-\frac{2}{3}$

Explanation:

Putting the first equation in slope-intercept form yields $y=\frac{3}{2}x+\frac{3}{2}$ .

A perpendicular line has a slope that is the negative inverse. In this case, $-\frac{2}{3}$ .

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Example Question #21 : How To Find The Equation Of A Perpendicular Line

Which of the following equations represents a line that goes through the point (-2, 6) and is perpendicular to the line 3x + 6y = 12 ?

Possible Answers:

y = 2x + 6

y = -2x + 10

y = 2x + 10

y = 2x + 2

y = -2x + 2

Correct answer:

y = 2x + 10

Explanation:

In order to solve this problem, we need first to transform the equation from standard form to slope-intercept form:

y = mx + b

Transform the original equation to find its slope.

3x + 6y = 12

First, subtract from both sides of the equation.

3x-3x + 6y = 12-3x

Simplify and rearrange.

6y = -3x + 12

Next, divide both sides of the equation by 6.

$\frac{6y}{6} = -\frac{3x}{6} + \frac{12}{6}$

$y = -\frac{1}{2}x+ 2$

The slope of our first line is equal to $-\frac{1}{2}$ . Perpendicular lines have slopes that are opposite reciprocals of each other; therefore, if the slope of one is x, then the slope of the other is equal to the following:

$-\frac{1}{x}$

Let's calculate the opposite reciprocal of our slope:

$-\frac{1}{2}\rightarrow -\left (-\frac{2}{1} \right )=2$

The slope of our line is equal to 2. We now have the following partial equation:

y = 2x + b

We are missing the y-intercept, . Substitute the x- and y-values in the given point (-2, 6) to solve for the missing y-intercept.

6 = 2(-2) + b

6 = -4 + b

Add 4 to both sides of the equation.

6+4 = -4+4 + b

10 = b

Substitute this value into our partial equation to construct the equation of our line:

y = 2x + 10

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Example Question #1 : How To Find The Midpoint Of A Line Segment

A line segment has endpoints (0,4) and (5,6). What are the coordinates of the midpoint?

Possible Answers:

(2.5,5)

(2.5,-5)

(3,9)

(0,6)

(0,4)

Correct answer:

(2.5,5)

Explanation:

A line segment has endpoints (0,4) and (5,6). To find the midpoint, use the midpoint formula:

X: (x₁+x₂)/2 = (0+5)/2 = 2.5

Y: (y₁+y₂)/2 = (4+6)/2 = 5

The coordinates of the midpoint are (2.5,5).

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Example Question #1 : How To Find The Midpoint Of A Line Segment

Find the midpoint between (-3,7) and (5,-9)

Possible Answers:

(-1,-1)

(4,-8)

(1,-8)

(4,-1)

(1,-1)

Correct answer:

(1,-1)

Explanation:

You can find the midpoint of each coordinate by averaging them. In other words, add the two x coordinates together and divide by 2 and add the two y coordinates together and divide by 2.

x-midpoint = (-3 + 5)/2 = 2/2 = 1

y-midpoint = (7 + -9)/2 = -2/2 = -1

(1,-1)

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Example Question #2 : How To Find The Midpoint Of A Line Segment

Find the coordinates for the midpoint of the line segment that spans from (1, 1) to (11, 11).

Possible Answers:

(5, 6)

(6, 5)

(5, 5)

(6, 6)

(7, 7)

Correct answer:

(6, 6)

Explanation:

The correct answer is (6, 6). The midpoint formula is ((x₁+ x₂)/2),((y₁+ y₂)/2) So 1 + 11 = 12, and 12/2 = 6 for both the x and y coordinates.

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Example Question #1 : How To Find The Midpoint Of A Line Segment

What is the midpoint between the points (–1, 2) and (3, –6)?

Possible Answers:

(1,2)

(3,1)

(1,–2)

(–1,2)

(–2,–1)

Correct answer:

(1,–2)

Explanation:

midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2)

= ((–1 + 3)/2, (2 – 6)/2)

= (2/2, –4/2)

= (1,–2)

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Example Question #1 : How To Find The Midpoint Of A Line Segment

A line segment connects the points $(-1,4)$ and $(3,16)$ . What is the midpoint of this segment?

Possible Answers:

$(-1,6)$

$(2,10)$

$(1,10)$

$(0,6)$

$(2,5)$

Correct answer:

$(1,10)$

Explanation:

To solve this problem you will need to use the midpoint formula:

$midpoint = (\frac{x_{1}+x_{2}}{2},\frac{y_{1}+y_{2}}{2} )$

Plug in the given values for the endpoints of the segment: $(-1,4)$ and $(3,16)$ .

$midpoint = (\frac{-1+3}{2},\frac{4+16}{2} ) = (\frac{2}{2}, \frac{20}{2}) = (1, 10)$

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Example Question #2 : How To Find The Midpoint Of A Line Segment

What is the midpoint between (1,4) and (5,12) ?

Possible Answers:

(2,10)

(4,8)

(3,8)

(3,6)

(8,2)

Correct answer:

(3,8)

Explanation:

The midpoint is the point halfway between the two endpoints, so sum up the coordinates and divide by 2:

$(\frac{1+5}{2},\frac{4+12}{2})=(\frac{6}{2},\frac{16}{2})=(3,8)$

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PSAT Math : Geometry

Study concepts, example questions & explanations for PSAT Math

All PSAT Math Resources

Example Questions

Example Question #1 : How To Find The Equation Of A Perpendicular Line

Example Question #2 : How To Find The Equation Of A Perpendicular Line

Example Question #1 : How To Find The Equation Of A Perpendicular Line

Example Question #21 : How To Find The Equation Of A Perpendicular Line

Example Question #1 : How To Find The Midpoint Of A Line Segment

Example Question #1 : How To Find The Midpoint Of A Line Segment

Example Question #2 : How To Find The Midpoint Of A Line Segment

Example Question #1 : How To Find The Midpoint Of A Line Segment

Example Question #1 : How To Find The Midpoint Of A Line Segment

Example Question #2 : How To Find The Midpoint Of A Line Segment

All PSAT Math Resources

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