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SAT Math : Geometry

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Example Questions

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

Line is given by $-\frac{6}{5}x+y=4$ .

Which of the following is perpendicular to ?

Possible Answers:

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

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

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

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

$\frac{3}{5}x+y=-2$

Correct answer:

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

Explanation:

Putting the equation of the line into slope-intercept form, we get

$y=\frac{6}{5}x+4$

The slope of line , therefore, is $\frac{6}{5}$ .

In order for a line to be perpendicular to the given line, it must have a slope that is the negative reciprocal of line g's slope.

The slope of any given line perpendicular to line g must be $-\frac{5}{6}$ when written in slope-intercept form. In other words, the equation of the perpendicular line must be $y=-\frac{5}{6}x +k$ where k is any constant.

Written in standard form, the equation of this perpendicular line must be $\frac{5}{6}x+y=k.$

Therefore, the most appropriate answer is $\frac{5}{6}x+y=6.$

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Example Question #2 : How To Find Out If Lines Are Perpendicular

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

The equation of Line A is y = 3x+ 17 .

The equation of Line B is -9x + 3y =17 .

The equation of Line is 6x - 2y = 17 .

Which of the following is a true statement?

Possible Answers:

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

None of the statements in the other choices is true.

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

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

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

Correct answer:

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

Explanation:

Two lines are perpendicular if and only if the product of their slopes is . Therefore, we need to find the slopes of all three lines.Rewrite the equation for each line in its slope-intercept form y = mx+ b , where is the slope of the line.

Line A:

y = 3x+ 17 is already in this form. The slope of Line A is the coefficient of , which is 3.

Line B:

-9x + 3y =17

Isolate by working the same operations on both sides:

-9x + 3y + 9x =17 + 9x

3y = 9x + 17

$\frac{1}{3} \cdot 3y =\frac{1}{3} \cdot ( 9x + 17)$

$y =\frac{1}{3} \cdot 9x +\frac{1}{3} \cdot 17$

$y =3 x +\frac{17}{3}$

The slope of Line B is the coefficient of , which is 3.

Line C:

6x - 2y = 17

6x - 2y - 6x= 17 - 6x

- 2y = - 6x + 17

$- \frac{1}{2}(- 2y) = - \frac{1}{2} (- 6x + 17)$

$y = - \frac{1}{2} (- 6x) + \left ( - \frac{1}{2} \right )(17)$

$y = 3x - \frac{17}{2}$

The slope of Line C is the coefficient of , which is 3.

All three lines have slope 3, so the product of the slopes of any two of the lines is $3 \times 3 = 9 \ne -1$ . Therefore, no two of the lines are perpendicular.

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

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

The equation of Line A is 2x+ 7y = 13 .

The equation of Line B is 7x - 2y = 13 .

The equation of Line C is 2x-7y = 13 .

Which of the following is a true statement?

Possible Answers:

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

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

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

None of the statements in the other choices is true.

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

Correct answer:

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

Explanation:

Two lines are perpendicular if and only if the product of their slopes is . Therefore, we need to find the slopes of all three lines.

Rewrite the equation for each line in its slope-intercept form y = mx+ b , where is the slope of the line.

Line A:

2x+ 7y = 13

Isolate by working the same operations on both sides:

2x+ 7y - 2x = 13 - 2x

7y = - 2x+ 13

$\frac{1}{7} \cdot 7y = \frac{1}{7} \cdot ( - 2x+ 13)$

$y = \frac{1}{7} \cdot ( - 2x) +\frac{1}{7} \cdot 13$

$y = - \frac{2}{7} x +\frac{13}{7}$

The slope of Line A is the coefficient of , which is $- \frac{2}{7}$ .

Take the same steps with the equations of the other two lines:

Line B:

7x - 2y = 13

7x - 2y -7x = 13-7x

- 2y = -7x + 13

$- \frac{1}{2}(- 2y) =- \frac{1}{2}( -7x + 13)$

$y = - \frac{1}{2}( -7x) +\left ( - \frac{1}{2} \right ) ( 13)$

$y = \frac{7}{2} x - \frac{13}{2}$

The slope of Line B is $\frac{7}{2}$

Line C:

2x-7y = 13

2x-7y - 2x = 13 - 2x

-7y = - 2x + 13

$-\frac{1}{7}(-7y) = -\frac{1}{7} (- 2x + 13)$

$y = -\frac{1}{7} (- 2x ) + \left ( -\frac{1}{7} \right )(13)$

$y = \frac{2}{7} x -\frac{13}{7}$

The slope of Line C is $\frac{2}{7}$ .

The product of the slopes of Lines A and B is $- \frac{2}{7} \times \frac{7}{2} = -1$ , so these two lines are perpendicular.

The product of the slopes of Lines A and C is $- \frac{2}{7} \times \frac{2}{7} = -\frac{4}{49}$ , so these two lines are not perpendicular.

The product of the slopes of Lines B and C is $\frac{7}{2} \times \frac{2}{7}= 1$ , so these two lines are not perpendicular.

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Example Question #7 : How To Find Out If Lines Are Perpendicular

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

Line A has intercepts (0,4) and (6,0 ) .

Line B has intercepts (0, -6) and (9, 0) .

Line C has intercepts (0,- 8 ) and (-12, 0 ) .

Which of the following statements is true?

Possible Answers:

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

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

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

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

None of the statements in the other choices is true.

Correct answer:

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

Explanation:

The slope of each line, given the coordinates of two points through which they pass, can be calculated by substituting the point coordinates into the slope formula

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

Line A:

Setting $y_{1} = x_{2}= 0, x_{1} = 6 , y_{2} = 4$ and substituting:

$m = \frac{4-0}{0- 6} = \frac{4}{-6} = -\frac{2}{3}$

Line B:

Setting $y_{1} = x_{2}= 0, x_{1} = 9 , y_{2} = -6$ and substituting:

$m = \frac{-6-0}{0- 9} = \frac{-6}{-9} = \frac{2}{3}$

Line C:

Setting $y_{1} = x_{2}= 0, x_{1} = -12 , y_{2} = -8$ and substituting:

$m = \frac{-8-0}{0- (-12)} = \frac{-8}{12} = -\frac{2}{3}$

The product of the slopes of Line A and Line B is $- \frac{2}{3} \times \frac{2}{3} = -\frac{4}{9}$ , as is the product of the slopes of Line C and Line B; Since neither product is equal to , neither pair of lines is perpendicular. Also, the slopes of Line A and Line C are equal, so the lines are parallel, not perpendicular.

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

Which of the following lines is perpendicular to $y=3x-4$

Possible Answers:

$y=-3x-4$

$y=\frac{1}{3}x-4$

$y=\frac{1}{3}x+4$

$y=-\frac{1}{3}x-4$

Correct answer:

$y=-\frac{1}{3}x-4$

Explanation:

The line which is perpendicular has a slope which is the negative inverse of the slope of the original line.

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Example Question #471 : Geometry

Trans

Lines P and Q are parallel. Find the value of .

Possible Answers:

$x=\pm \sqrt{ 205}$

$x=5 + \sqrt{ 205}$

$x= \sqrt{ 205}$

$x=-5+ \sqrt{ 205}$

$x=-5- \sqrt{ 205}$

Correct answer:

$x=-5+ \sqrt{ 205}$

Explanation:

Since these are complementary angles, we can set up the following equation.

x^2+4+10x-4=180

x^2+10x=180

x^2+10x-180=0

Now we will use the quadratic formula to solve for .

$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

$x=\frac{-10\pm \sqrt{10^2-4(1)(-180)}}{2(1)}$

$x=\frac{-10\pm \sqrt{100+720}}{2}$

$x=\frac{-10\pm \sqrt{820}}{2}$

$x=\frac{-10\pm \sqrt{4\cdot 205}}{2}$

$x=\frac{-10\pm 2\sqrt{ 205}}{2}$

$x=-5\pm \sqrt{ 205}$

Note, however, that the measure of an angle cannot be negative, so $x=-5- \sqrt{ 205}$ is not a viable answer. The correct answer, then, is $x=-5+ \sqrt{ 205}$

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Example Question #82 : Coordinate Geometry

In the xy -plane, line l is given by the equation 2x - 3y = 5. If line l passes through the point (a ,1), what is the value of a ?

Possible Answers:

-1

-2

Correct answer: 4

Explanation:

The equation of line l relates x -values and y -values that lie along the line. The question is asking for the x -value of a point on the line whose y -value is 1, so we are looking for the x -value on the line when the y-value is 1. In the equation of the line, plug 1 in for y and solve for x:

2x - 3(1) = 5

2x - 3 = 5

2x = 8

x = 4. So the missing x-value on line l is 4.

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Example Question #91 : Lines

The equation of a line is: 2x + 9y = 71

Which of these points is on that line?

Possible Answers:

(-2,7)

(4,7)

(4,-7)

(-4,7)

(2,7)

Correct answer:

(4,7)

Explanation:

Test the difference combinations out starting with the most repeated number. In this case, y = 7 appears most often in the answers. Plug in y=7 and solve for x. If the answer does not appear on the list, solve for the next most common coordinate.

2(x) + 9(7) = 71

2x + 63 = 71

2x = 8

x = 4

Therefore the answer is (4, 7)

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Example Question #24 : Psat Mathematics

Which of the following lines contains the point (8, 9)?

Possible Answers:

$\dpi{100} \small 8x=9y$

$\dpi{100} \small 3x+6=2y$

$\dpi{100} \small 3x+6=y$

$\dpi{100} \small 8x+9=y$

$\dpi{100} \small 3x-6=2y$

Correct answer:

$\dpi{100} \small 3x-6=2y$

Explanation:

In order to find out which of these lines is correct, we simply plug in the values $\dpi{100} \small x=8$ and $\dpi{100} \small y=9$ into each equation and see if it balances.

The only one for which this will work is $\dpi{100} \small 3x-6=2y$

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Example Question #472 : Geometry

$\dpi{100} \small 5x+25y = 125$

Which point lies on this line?

Possible Answers:

$\dpi{100} \small (5,5)$

$\dpi{100} \small (1,5)$

$\dpi{100} \small (5,1)$

$\dpi{100} \small (1,4)$

$\dpi{100} \small (5,4)$

Correct answer:

$\dpi{100} \small (5,4)$

Explanation:

$\dpi{100} \small 5x+25y = 125$

Test the coordinates to find the ordered pair that makes the equation of the line true:

$\dpi{100} \small (5,4)$

$\dpi{100} \small 5 (5) + 25 (4) = 25 + 100 = 125$

$\dpi{100} \small (1,5)$

$\dpi{100} \small 5(1)+25(5)= 5+125=130$

$\dpi{100} \small (5,1)$

$\dpi{100} \small 5(5)+25(1)= 25+25=50$

$\dpi{100} \small (5,5)$

$\dpi{100} \small 5(5)+25(5)= 25+125=150$

$\dpi{100} \small (1,4)$

$\dpi{100} \small 5(1)+25(4)= 5+100=105$

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SAT Math : Geometry

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

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

Example Question #2 : How To Find Out If Lines Are Perpendicular

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

Example Question #7 : How To Find Out If Lines Are Perpendicular

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

Example Question #471 : Geometry

Example Question #82 : Coordinate Geometry

Example Question #91 : Lines

Example Question #24 : Psat Mathematics

Example Question #472 : Geometry

All SAT Math Resources

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