# SAT Math : Coordinate Geometry

## Example Questions

### Example Question #3 : How To Find Out If Lines Are Parallel

Which of the following lines is parallel to ?

Explanation:

The line presented in the question is . Parallel lines have the same slope. The slope of the presented equation is . Therefore, the answer is:

### Example Question #4 : How To Find Out If Lines Are Parallel

is parallel to which of the following lines?

Explanation:

Remember that when a line is stated as , its slope-intercept conversion has a slope of .

Also bear in mind that we are looking for a parallel line, which means we need to find another line where .

is our answer because , which is what we are looking for.

### Example Question #31 : Coordinate Geometry

Which of the following are parallel to the line ?

A.

B.

C.

D.

E.

A & D

A, B, & C

C & D

D & E

A & D

Explanation:

In order for two lines to be parallel, they must have the same slope and different y-intercepts. In slope-intercept form, the slope of the coefficient of our  value.

We want to find lines that have a slope of . The two answers that share this slope with the given equation are  and , which correspond with answers A and D.

### Example Question #32 : Coordinate Geometry

Given lines on the coordinate plane as follows:

Line A has equation

Line B has equation

Line C has equation

Which of the following is a true statement?

Line A and Line B are parallel to each other, but Line C is parallel to neither Line A nor Line B.

No two of Line A, Line B, and Line C are parallel to each other.

Line A and Line C are parallel to each other, but Line B is parallel to neither Line A nor Line C.

Line B and Line C are parallel to each other, but Line A is parallel to neither Line B nor Line C.

Line A, Line B, and Line C are parallel to one other.

No two of Line A, Line B, and Line C are parallel to each other.

Explanation:

Two lines are parallel if and only if they have the same slope. Therefore, we must find the slope of each line. We do this by rewriting each equation in slope-intercept form , with -coefficient  being the slope of the line.

In each case, solve for  by isolating this variable on the left side.

Line A:

The slope of Line A is the -coefficient .

Line B:

The slope of Line Bis the -coefficient .

Line C:

The slope of Line C is the -coefficient .

No two of the given lines have the same slope, so no two are parallel.

### Example Question #33 : Coordinate Geometry

Given Lines A, B, and C on the coordinate plane as follows:

The equation of Line A is .

The equation of Line B is .

The equation of Line C is .

Which of the following is a true statement?

Line A and Line C are parallel to each other, but Line B is parallel to neither Line A nor Line C.

Line A, Line B, and Line C are parallel to one other.

No two of Line A, Line B, and Line C are parallel to each other.

Line B and Line C are parallel to each other, but Line A is parallel to neither Line B nor Line C.

Line A and Line B are parallel to each other, but Line C is parallel to neither Line A nor Line B.

Line A, Line B, and Line C are parallel to one other.

Explanation:

Two lines are parallel if and only if they have the same slope. Therefore, we must find the slope of each line. We do this by rewriting each equation in slope-intercept form , with -coefficient  being the slope of the line.

In each case, solve for  by isolating this variable on the left side.

Line A:

.

This equation is already in slope-intercept form. The slope of Line A is the -coefficient .

Line B:

The slope of Line B is the -coefficient .

Line C:

The slope of Line C is the -coefficient .

In each case, the slope is . The three lines, having the same slope, are all parallel to one another.

### Example Question #34 : Coordinate Geometry

Given Lines A, B, and C on the coordinate plane as follows:

Line A has intercepts  and .

Line B has intercepts  and .

Line C has intercepts  and .

Which statement is true?

Lines A and C are parallel to each other, but Line B is parallel to neither line.

Lines A, B, and C are all parallel to one another.

Lines B and C are parallel to each other, but Line A is parallel to neither line.

Lines A and B are parallel to each other, but Line C is parallel to neither line.

No two of Lines A, B, and C are parallel.

Lines A and C are parallel to each other, but Line B is parallel to neither line.

Explanation:

Two lines are parallel if an only if their slopes are equal. The slope of a line with -intercept  and -intercept  can be determined using the formula

.

Calculate the slope of Line A by setting :

Calculate the slope of Line B by setting :

Calculate the slope of Line B by setting :

Lines A and C have the same slope and are parallel; Line B has a different slope and is not parallel to the other two.

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

Which of the following lines is parallel to:

Explanation:

First write the equation in slope intercept form. Add  to both sides to get . Now divide both sides by  to get . The slope of this line is , so any line that also has a slope of  would be parallel to it. The correct answer is  .

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

The equation of line p is  y= 1/4x +6.  If line k contains the point (3,5) and is perpendicular to line p, find the equation of line k.

y = 3x + 5

y = 4x - 17

y = 1/4x + 17

y = -4x + 17

y = -4x + 17

Explanation:

Using the slope intercept formula, we can see the slope of line p is ¼. Since line k is perpendicular to line p it must have a slope that is the negative reciprocal. (-4/1) If we set up the formula y=mx+b, using the given point and a slope of (-4), we can solve for our b or y-intercept. In this case it would be 17.

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

In the xy-coordinate plane, a line A contains the points (0,0) and (3,1). If the line B is perpendicular to A at (3,1), what is the equation of the line?

y = -1/3x + 10

y = -3x + 1

y = 1/3x + 1

y = -3x + 10

y = 3x + 1

y = -3x + 10

Explanation:

First, you need to obtain the equation of the first line, A. Its slope is given by:

(y2 - y1) / (x2 - x1) = (1 - 0) / (3 - 0) = 1/3 = slope of A.

Remember that the slope of a perpendicular line to a given line is -1 times the inverse of its slope. Thus the slope of B:

(-1) x 1 / (1/3) = -3

Thus with y = mx + b, m = -3. Now the line must include (3,1). Thus:

with y = -3x + b:

1 = -3(3) + b;

1 = -9 + b; add 9 to both sides:

10 = b

### Example Question #3 : How To Find The Equation Of A Perpendicular Line

What line is perpendicular to the line 2x + 3y = 6 through (4, 1)?

–2x + 3y = 5

3x + 2y = 10

2x + 3y = –5

–3x + 2y = –10

–x + 4y = 8