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# SSAT Upper Level Math : Properties of Parallel and Perpendicular Lines

## Example Questions

### Example Question #1 : Properties Of Parallel And Perpendicular Lines

Line P passes through the origin and point .

Line Q passes through the origin and point .

Line R passes through the origin and point .

Line S passes through the origin and point .

Which of these lines is parallel to the line of the equation  ?

Line P

Line R

Line S

None of the other responses is correct.

Line Q

Line S

Explanation:

First, find the slope of the line of the equation  by rewriting it in slope-intercept form:

The slope of this line is , so we are looking for a line which also has this slope.

Find the slopes of all four lines by using the slope formula . Since each line passes through the origin, this formula can be simplified to

using the other point.

Line P:

Line Q:

Line R:

Line S:

Line S has the desired slope and is the correct choice.

### Example Question #2 : Properties Of Parallel And Perpendicular Lines

You are given three lines as follows:

Line A includes points  and .

Line B includes point  and has -intercept .

Line C includes the origin and point .

Which lines are parallel?

Explanation:

Find the slope of all three lines using the slope formula :

Line A:

Line B:

Line C:

Lines A and C have the same slope; Line B has a different slope. Only Lines A and C are parallel.

### Example Question #3 : Properties Of Parallel And Perpendicular Lines

Line A has equation .

Line B has equation .

Which statement is true of the two lines?

Explanation:

Write each statement in slope-intercept form:

Line A:

The slope is .

Line B:

The slope is .

The lines have differing slopes, so they are neither identical nor parallel. The product of the slopes is , so they are not perpendicular. The correct response is that they are distinct lines that are neither parallel nor perpendicular.

### Example Question #4 : Properties Of Parallel And Perpendicular Lines

Three lines are drawn on the coordinate plane.

The green line has slope , and -intercept .

The blue line has slope , and  -intercept .

The red line has slope , and -intercept .

Which two lines are perpendicular to each other?

The blue line and the green line are perpendicular.

No two of these lines are perpendicular.

It cannot be determined from the information given.

The blue line and the red line are perpendicular.

The green line and the red line are perpendicular.

The blue line and the red line are perpendicular.

Explanation:

To demonstrate two perpendicular lines, multiply their slopes; if their product is , then the lines are perpendicular (the -intercepts are irrelevant).

The products of these lines are given here.

Blue and green lines:

Red and green lines:

Blue and red lines:

It is the blue and red lines that are perpendicular.

We can also see that their slopes are negative reciprocals, indicating perpendicular lines.

### Example Question #5 : Properties Of Parallel And Perpendicular Lines

Two perpendicular lines intersect at point . One line also includes point . What is the slope of the other line?

Insufficient information is given to answer the question.

Explanation:

The slopes of two perpendicular lines are the opposites of each other's reciprocals.

To find the slope of the first line substitute  in the slope formula:

The slope of the first line is , so the slope of the second line is the opposite reciprocal of this, which is .

### Example Question #6 : Properties Of Parallel And Perpendicular Lines

Two perpendicular lines intersect at the origin; one line also passes through point . What is the slope of the other line?

Insufficient information is given to solve the problem.

Explanation:

The slopes of two perpendicular lines are the opposites of each other's reciprocals.

To find the slope of the first line, substitute  in the slope formula:

The slope of the first line is , so the slope of the second line is the opposite reciprocal of this, which is .

### Example Question #7 : Properties Of Parallel And Perpendicular Lines

Which of the following lines is perpendicular to the line ?

Explanation:

All we care about for this problem is the slopes of the lines...the x- and y-intercepts are irrelevant.

Remember that the slopes of perpendicular lines are opposite reciprocals. By putting the given equation into  form, we can see that its slope is . So we are looking for a line with a slope of .

The equation  can be put into the form , and so we know that it is perpendicular to the given line.

### Example Question #8 : Properties Of Parallel And Perpendicular Lines

Line A passes through the origin and .

Line B passes through the origin and .

Line C passes through the origin and .

Line D passes through the origin and .

Line E passes through the origin and .

Which line is perpendicular to Line A?

None of the other lines is perpendicular to A.

Line E

Line C

Line D

Line B

Line D

Explanation:

Find the slopes of all five lines using the slope formula . Since each line passes through the origin, this formula can be simplified to

using the other point.

Line A:

The correct line must have as its slope the opposite of the reciprocal of this, which is .

Line B:

Line C:

Line D:

Line E:

Of the last four lines, only Line D has the desired slope.

### Example Question #9 : Properties Of Parallel And Perpendicular Lines

Line W passes through the origin and point .

Line X passes through the origin and point .

Line Y passes through the origin and point .

Line Z passes through the origin and point .

Which of these lines is perpendicular to the line of the equation  ?

Line X

None of the other responses is correct.

Line Y

Line Z

Line W

Line Z

Explanation:

First, find the slope of the line of the equation  by rewriting it in slope-intercept form:

The slope of this line is , so we are looking for a line whose slope is the opposite of the reciprocal of this, or

Find the slopes of all four lines by using the slope formula . Since each line passes through the origin, this formula can be simplified to

using the other point.

Line W:

Line X:

Line Y:

Line Z:

Line Z has the desired slope and is the correct choice.