GMAT Math : Geometry

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Example Questions

Example Question #2 : Calculating The Equation Of A Line

Determine the equation of a line that has the points (-2,3) and (6,7) ?

Possible Answers:

y=-2x-2

$y=\frac{1}{2}x+4$

y=2x+2

$y=-\frac{1}{2}x-4$

$y=-\frac{1}{4}x+2$

Correct answer:

$y=\frac{1}{2}x+4$

Explanation:

The equation for a line in standard form is written as follows:

y=mx+b

Where is the slope and is the y intercept. We start by calculating the slope between the two given points using the following formula:

$m=\frac{y_2-y_1}{x_2-x_1}=\frac{7-3}{6-(-2)}=\frac{4}{8}=\frac{1}{2}$

Now we can plug either of the given points into the formula for a line with the calculated slope and solve for the y intercept:

y=mx+b

$3=(\frac{1}{2})(-2)+b$

b=4

We now have the slope and the y intercept of the line, which is all we need to write its equation in standard form:

$y=\frac{1}{2}x+4$

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Example Question #7 : Calculating The Equation Of A Line

Give the equation of the line that passes through the -intercept and the vertex of the parabola of the equation

$y = x^{2} + 5x - 6$ .

Possible Answers:

y=6x-6

$y=\frac{5}{2 } x - 6$

y = -6x+6

$y=-\frac{5}{2 } x + 6$

y=-x-6

Correct answer:

$y=\frac{5}{2 } x - 6$

Explanation:

The -intercept of the parabola of the equation can be found by substituting 0 for :

$y = x^{2} + 5x - 6$

$y = 0^{2} + 5 (0) - 6$

y = -6

This point is (0, -6) .

The vertex of the parabola of the equation $y = a x^{2} + bx+c$ has -coordinate $- \frac{b}{2a}$ , and its -coordinate can be found using substitution for . Setting a = 1 and b= 5 :

$x = - \frac{b}{2a} = - \frac{5}{2 \cdot 1} = - \frac{5}{2 }$

$y = \left ( - \frac{5}{2 } \right )^{2} + 5 \left ( - \frac{5}{2 } \right ) - 6$

$y = \frac{25}{4 } - \frac{25}{2 } - 6$

$y = \frac{25}{4 } - \frac{50}{4 } - \frac{24}{4}$

$y = - \frac{49}{4 }$

The vertex is $\left (- \frac{5}{2 },- \frac{49}{4 } \right )$

The line connects the points (0, -6) and $\left (- \frac{5}{2 },- \frac{49}{4 } \right )$ . Its slope is

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

$= \frac{-6-\left ( - \frac{49}{4 } \right )}{0 - \left (- \frac{5}{2 } \right )}$

$= \frac{-\frac{24 }{4 } + \frac{49}{4 } }{ \frac{5}{2 } }$

$= \frac{ \frac{25}{4 } }{ \frac{5}{2 } }$

$= \frac{25}{4 } \cdot \frac{2 } {5}$

$=\frac{5}{2 }$

Since the line has -intercept (0, -6) and slope $m =\frac{5}{2 }$ , the equation of the line is $y=\frac{5}{2 } x + (- 6)$ , or $y=\frac{5}{2 } x - 6$ .

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Example Question #681 : Geometry

What is the slope of the line Ax + 2Ay = 3A ?

Possible Answers:

$-\frac{A}{2}$

$-\frac{1}{2}$

$-\frac{2}{A}$

$\frac{A}{2}$

Correct answer:

$-\frac{1}{2}$

Explanation:

Rewrite this equation in slope-intercept form: y = mx + b , where is the slope.

Ax + 2Ay = 3A

2Ay = - Ax + 3A

$\frac{2Ay}{2A} = \frac{- Ax + 3A}{2A}$

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

The slope is the coefficient of , which is $-\frac{1}{2}$ .

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Example Question #682 : Geometry

Give the slope of the line of the equation: 0.4x - 0.5y = 20

Possible Answers:

$-\frac{5}{4}$

$-\frac{4}{5}$

$\frac{4}{5}$

$\frac{5}{4}$

Correct answer:

$\frac{4}{5}$

Explanation:

Rewrite in the slope-intercept form y = mx + b :

0.4x - 0.5y = 20

$\left (0.4x - 0.5y \right )\cdot 10 = 20 \cdot 10$

$\left (0.4x \right )\cdot 10-\left ( 0.5y \right )\cdot 10 = 200$

4x-5y = 200

4x-4x-5y = 200-4x

-5y = -4x+ 200

$-5y \div (-5)=\left ( -4x+ 200 \right )\div (-5)$

$y=\left ( -4x \right ) \div (-5)+\left ( 200 \right ) \div (-5)$

$y= \frac{4}{5}x-40$

The slope is the coefficient of , which is $\frac{4}{5}$ .

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Example Question #1 : Calculating The Slope Of A Line

Give the slope of the line of the equation: $\frac{}{}$ $\frac{2}{5}x + \frac{6}{7}y = 9$

Possible Answers:

$-\frac{7}{15}$

$\frac{21}{2}$

$\frac{45}{2}$

$-\frac{15}{7}$

$\frac{12}{35}$

Correct answer:

$-\frac{7}{15}$

Explanation:

Rewrite in the slope-intercept form y = mx + b :

$\frac{2}{5}x + \frac{6}{7}y = 9$

$\frac{2}{5}x - \frac{2}{5}x + \frac{6}{7}y = - \frac{2}{5}x+ 9$

$\frac{6}{7}y = - \frac{2}{5}x+ 9$

$\frac{6}{7}y \cdot \frac{7}{6}=\left ( - \frac{2}{5}x+ 9 \right )\cdot \frac{7}{6}$

$y=- \frac{2}{5}\cdot \frac{7}{6}x+ 9 \cdot \frac{7}{6}$

$y=-\frac{7}{15}x+ \frac{21}{2}$

The slope is the coefficient of , which is $-\frac{7}{15}$

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Example Question #681 : Geometry

Give the slope of the line of the equation

5y = x - 7

Possible Answers:

$-\frac{1}{5}$

$\frac{7}{5}$

$-\frac{7}{5}$

$\frac{1}{5}$

Correct answer:

$\frac{1}{5}$

Explanation:

Rewrite in the slope-intercept form y = mx + b :

5y = x - 7

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

$y = \frac{1}{5} x - \frac{7}{5} \right )$

The slope is the coefficient of , or $\frac{1}{5}$ .

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Example Question #2 : Calculating The Slope Of A Line

What is the slope of the line that contains (1,4) and (7,10) ?

Possible Answers:

$\frac{3}{2}$

$\frac{2}{3}$

$\frac{14}{8}$

Correct answer:

Explanation:

The slope formula is:

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

$(1,4)\ and\ (7,10)$

$m= \frac{7-1}{10-4}$

$m= \frac{6}{}6$

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Example Question #683 : Geometry

What is the slope of the line that contains (1,2) and (5,2) ?

Possible Answers:

$\frac{4}{6}$

$\frac{6}{4}$

$\frac{1}{4}$

Correct answer:

Explanation:

The slope formula is:

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

$(1,2)\ and\ (5,2)$

$m=\frac{2-2}{5-1}$

$m=\frac{0}{4}$

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Example Question #11 : Other Lines

What is the slope of the line that contains (-2,-1) and (-8,7) ?

Possible Answers:

$-\frac{3}{4}$

$\frac{3}{4}$

$-\frac{4}{3}$

$\frac{4}{3}$

Correct answer:

$-\frac{4}{3}$

Explanation:

The slope formula is:

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

$(-2,-1)\ and\ (-8,7)$

$m=\frac{7-(-1)}{-8-(-2)}$

$m=\frac{7+1}{-8+2}$

$m=\frac{8}{-6}$

$m=-\frac{4}{3}$

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Example Question #3 : Calculating The Slope Of A Line

A iine goes through points (6,N + 4 ) and (3,2N + 6) . What is its slope?

Possible Answers:

$\frac{3}{N+2}$

$\frac{N+2}{3}$

$\frac{N-2}{3}$

$\frac{-N-2}{3}$

$\frac{3}{N-2}$

Correct answer:

$\frac{-N-2}{3}$

Explanation:

Substitute x_1= 6 , y_1=N + 4,x_2= 3, y_2 =2N + 6 in the slope formula:

$m = \frac{y_2-y_1}{x_2-x_1}$

$m = \frac{(2N+6 )- (N+4)}{3-6} = \frac{2N+6 -N-4}{-3}= \frac{N+2}{-3} = \frac{-N-2}{3}$

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GMAT Math : Geometry

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #2 : Calculating The Equation Of A Line

Example Question #7 : Calculating The Equation Of A Line

Example Question #681 : Geometry

Example Question #682 : Geometry

Example Question #1 : Calculating The Slope Of A Line

Example Question #681 : Geometry

Example Question #2 : Calculating The Slope Of A Line

Example Question #683 : Geometry

Example Question #11 : Other Lines

Example Question #3 : Calculating The Slope Of A Line

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