Properties of Parallel and Perpendicular Lines

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SSAT Upper Level Quantitative › Properties of Parallel and Perpendicular Lines

Questions 1 - 10

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

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

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

and

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

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

Explanation

All of the equations are given in slope-intercept form , 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

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.

One side of a rectangle on the coordinate plane has as its endpoints the points and .

What would be the slope of a side adjacent to this side?

$\frac{7}{5}$

$\frac{5}{7}$

$\frac{3}{7}$

$\frac{7}{3}$

None of the other responses gives the correct answer.

Explanation

First, we find the slope of the segment connecting or . Using the formula

$m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

and setting

$x_{1} = 2, y_{1}=4, x_{2}= 9, y_{2} = -1$

we get

$m = \frac{-1-4}{9-2}= \frac{-5}{7} = -\frac{5}{7}$

Adjacent sides of a rectangle are perpendicuar, so their slopes will be the opposites of each other's reciprocals. Therefore, the slope of an adjacent side will be the opposite of the reciprocal of $-\frac{5}{7}$ , which is $\frac{7}{5}$ .

One side of a rectangle on the coordinate plane has as its endpoints the points and .

What would be the slope of a side adjacent to this side?

$\frac{7}{5}$

$\frac{5}{7}$

$\frac{3}{7}$

$\frac{7}{3}$

None of the other responses gives the correct answer.

Explanation

First, we find the slope of the segment connecting or . Using the formula

$m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

and setting

$x_{1} = 2, y_{1}=4, x_{2}= 9, y_{2} = -1$

we get

$m = \frac{-1-4}{9-2}= \frac{-5}{7} = -\frac{5}{7}$

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

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

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

and

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

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

Explanation

Of the five pairs, only

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.

Line A has equation .

Line B has equation .

Which statement is true of the two lines?

$\text{The lines are different but neither parallel nor perpendicular.}$

$\text{The lines are parallel.}$

$\text{The lines are perpendicular.}$

$\text{The lines are identical.}$

$\text{None of the other four statements are true.}$

Explanation

Write each statement in slope-intercept form:

Line A:

$y =-\frac{1}{7}\left ( -5x+ 20 \right )$

$y = \frac{5}{7} x-\frac{ 20}{7}$

The slope is $\frac{5}{7}$ .

Line B:

$y = -\frac{1}{5 }\left ( - 7x+ 32 \right )$

$y = \frac{7}{5}x-\frac{32}{5}$

The slope is $\frac{7}{5}$ .

The lines have differing slopes, so they are neither identical nor parallel. The product of the slopes is $\frac{5}{7} \times \frac{7}{5} = 1 e -1$ , so they are not perpendicular. The correct response is that they are distinct lines that are neither parallel nor perpendicular.

Parallel

Figure NOT drawn to scale

In the above figure, . Evaluate .

Explanation

The two marked angles are same-side exterior angles of two parallel lines formed by a transversal ,; by the Parallel Postulate, the angles are supplementary - the sum of their measures is 180 degrees. Therefore,

Parallel

Figure NOT drawn to scale

In the above figure, . Express in terms of .

Explanation

The two marked angles are same-side interior angles of two parallel lines formed by a transversal ; by the Parallel Postulate, the angles are supplementary - the sum of their measures is 180 degrees. Therefore,

Solve for by moving the other terms to the other side and simplifying:

Parallel

Figure NOT drawn to scale

In the above figure, . Express in terms of .

Explanation

Solve for by moving the other terms to the other side and simplifying:

Line A has equation .

Line B has equation .

Which statement is true of the two lines?

$\text{The lines are different but neither parallel nor perpendicular.}$

$\text{The lines are parallel.}$

$\text{The lines are perpendicular.}$

$\text{The lines are identical.}$

$\text{None of the other four statements are true.}$

Explanation

Write each statement in slope-intercept form:

Line A:

$y =-\frac{1}{7}\left ( -5x+ 20 \right )$

$y = \frac{5}{7} x-\frac{ 20}{7}$

The slope is $\frac{5}{7}$ .

Line B:

$y = -\frac{1}{5 }\left ( - 7x+ 32 \right )$

$y = \frac{7}{5}x-\frac{32}{5}$

The slope is $\frac{7}{5}$ .

Parallel

Figure NOT drawn to scale

In the above figure, . Evaluate .

Explanation

Angles of degree measures and form a linear pair, making the angles supplementary - that is, their degree measures total 180. Therefore,

Solving for :

$4y \div 4 = 192 \div 4$

The angles of measures and form a pair of alternating interior angles of parallel lines, so, as a consequence of the Parallel Postulate, they are congruent, and

Substituting for :

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