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SSAT Upper Level Math : How to find whether lines are perpendicular

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Example Questions

Example Question #11 : How To Find Whether Lines Are Perpendicular

Lines 1 and 2, which are perpendicular, have their -intercepts at the point (0, 8) . The -intercept of Line 1 is at the point (5,0) . Give the -intercept of Line 2.

Possible Answers:

$\left ( \frac{5 }{64}, 0\right )$

$\left ( 12 \frac{4}{5}, 0 \right )$

$\left ( -12 \frac{4}{5}, 0 \right )$

$\left (-\frac{5 }{64}, 0\right )$

(-5,0)

Correct answer:

$\left ( -12 \frac{4}{5}, 0 \right )$

Explanation:

The slope of a line with -intercept (a, 0) and -intercept (0, b) is $m = -\frac {b}{a}$ . For Line 1, b= 8, a= 5 , so Line 1 has slope $-\frac{8}{5}$ . The slope of Line 2, which is perpendicular to Line 1, will be the opposite of the reciprocal of this, which is $\frac{5}{8}$ . Setting equal to this and b=8 , we get

$\frac{5}{8}= -\frac {8}{a}$ , or $\frac{5}{8}= \frac {-8}{a}$

Cross-multiplying:

5a = -64

$\frac{5a }{5}= \frac{-64}{5}$

$a = -12 \frac{4}{5}$

The -intercept of Line 2 is $\left ( -12 \frac{4}{5}, 0 \right )$ .

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Example Question #12 : How To Find Whether Lines Are Perpendicular

Which of the following choices gives the equations of a pair of perpendicular lines with the same -intercept?

Possible Answers:

y = 2x+ 6 and $y = \frac{1}{2}x + 6$

y = 4x-8 and $y = - \frac{1}{4}x + \frac{1}{8}$

$y = \frac{2}{5}x + 9$ and $y = - \frac{5}{2}x + 9$

y = 3x+ 7 and y = -3x + 7

$y = \frac{1}{3}x + \frac{2}{3}$ and $y = - \frac{1}{3}x - \frac{2}{3}$

Correct answer:

$y = \frac{2}{5}x + 9$ and $y = - \frac{5}{2}x + 9$

Explanation:

All of the equations are given in slope-intercept form y = mx+b , so we can answer this question by examining the coefficients of , which are the slopes, and the constants, which are the -intercepts. In each case, since the lines are perpendicular, each -coefficient must be the other's opposite reciprocal, and since the lines have the same -intercept, the constants must be equal.

Of the five pairs, only

y = 4x-8 and $y = - \frac{1}{4}x + \frac{1}{8}$

and

$y = \frac{2}{5}x + 9$ and $y = - \frac{5}{2}x + 9$

have equations whose -coefficients are the other's opposite reciprocal. Of these, only the latter pair of equations have equal constant terms.

$y = \frac{2}{5}x + 9$ and $y = - \frac{5}{2}x + 9$

is the correct choice.

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Example Question #13 : How To Find Whether Lines Are Perpendicular

Given: the following three lines on the coordinate plane:

Line 1: The line of the equation x+2y = 12

Line 2: The line of the equation 2x+y= 12

Line 3: The line of the equation y=2 x

Which of the following is a true statement?

Possible Answers:

None of the other responses is correct.

No two of Line 1, Line 2, or Line 3 form a pair of perpendicular lines.

Line 1 and Line 3 are perpendicular; Line 2 is perpendicular to neither.

Line 2 and Line 3 are perpendicular; Line 1 is perpendicular to neither.

Line 1 and Line 2 are perpendicular; Line 3 is perpendicular to neither.

Correct answer:

Line 1 and Line 3 are perpendicular; Line 2 is perpendicular to neither.

Explanation:

The slope of each line can be calculated by putting the equation in slope-intercept form y = mx+b and noting the coefficient of :

Line 1:

x+2y = 12

x+2y - x = 12 - x

2y = -1x + 12

$\frac{2y }{2}= \frac{-1x + 12}{2}$

$y= -\frac{1}{2}x+6$

Slope of Line 1: $-\frac{1}{2}$

Line 2:

2x+y= 12

2x+y - 2x= 12 - 2x

y = -2x+12

Slope of Line 2:

Line 3: The equation is already in slope-intercept form; its slope is 2.

Two lines are perpendicular if and only their slopes have product . The slopes of Lines 1 and 3 have product $-\frac{1}{2} \cdot 2 = -1$ ; they are perpendicular. The slopes of Lines 1 and 2 have product $-\frac{1}{2} \cdot (- 2 )= 1$ ; they are not perpendicular. The slopes of Lines 2 and 3 have product $-2 \cdot 2 = -4$ ; they are not perpendicular.

Correct response: Line 1 and Line 3 are perpendicular; Line 2 is perpendicular to neither.

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SSAT Upper Level Math : How to find whether lines are perpendicular

Study concepts, example questions & explanations for SSAT Upper Level Math

All SSAT Upper Level Math Resources

Example Questions

Example Question #11 : How To Find Whether Lines Are Perpendicular

Example Question #12 : How To Find Whether Lines Are Perpendicular

Example Question #13 : How To Find Whether Lines Are Perpendicular

All SSAT Upper Level Math Resources

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