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Intermediate Geometry : Intermediate Geometry

Study concepts, example questions & explanations for Intermediate Geometry

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Example Questions

Example Question #143 : Lines

Find the slope of the line that passes through the points:

(92, 14) and (100, 20)

Possible Answers:

$-\frac{1}{4}$

$\frac{3}{4}$

$\frac{1}{2}$

Correct answer:

$\frac{3}{4}$

Explanation:

Recall that the slope of a line also measures how steep the line is. Use the following equation to find the slope of the line:

$\text{Slope}=\frac{y_2-y_1}{x_2-x_1}$

Now, substitute in the information using the given points.

$\text{Slope}=\frac{20-14}{100-92}$

Simplify.

$\text{Slope}=\frac{6}{8}$

Solve.

$\text{Slope}=\frac{3}{4}$

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Example Question #72 : Other Lines

Find the slope of the line that passes through the points:

(-10, -14) and (34, 12)

Possible Answers:

$-\frac{7}{11}$

$\frac{5}{11}$

$\frac{13}{22}$

$\frac{1}{2}$

Correct answer:

$\frac{13}{22}$

Explanation:

Recall that the slope of a line also measures how steep the line is. Use the following equation to find the slope of the line:

$\text{Slope}=\frac{y_2-y_1}{x_2-x_1}$

Now, substitute in the information using the given points.

$\text{Slope}=\frac{12-(-14)}{34-(-10)}$

Simplify.

$\text{Slope}=\frac{26}{44}$

Solve.

$\text{Slope}=\frac{13}{22}$

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Example Question #1431 : Intermediate Geometry

What is the slope m of the line below?

y - 2= 2(x-4)

Possible Answers:

m=-2

m = 2

m=4

m=-4

m=-8

Correct answer:

m = 2

Explanation:

To answer this question you should realize that the equation for the line is given in point - slope form. The standard point slope form of a line is given below:

y-y_1 = m(x-x_1)

m represents the slope of the line so all we have to do is recognize the correct line form and we automatically know that the slope is m = 2

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Example Question #141 : Coordinate Geometry

Give the slope of the line of the equation 8y = 1

Possible Answers:

$\textup{The slope is undefined}$

$\frac{1}{8}$

Correct answer:

Explanation:

If we divide both sides by 8, as follows:

8y = 1

$\frac{8y}{8} = \frac{1}{8}$

$y = \frac{1}{8}$

we see that the equation is equivalent to one of the form y = b . This is a horizontal line, and its slope is 0.

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Example Question #141 : Coordinate Geometry

Give the slope of the line of the equation $- \frac{2}{3}x = 4$ .

Possible Answers:

$\textup{The slope is undefined}$

$- \frac{3}{2}$

$- \frac{2}{3}$

Correct answer:

$\textup{The slope is undefined}$

Explanation:

If we multiply both sides of the equation by $-\frac{3}{2}$ , as follows:

$- \frac{2}{3}x = 4$

$- \frac{3}{2} \cdot \left (- \frac{2}{3}x \right )=- \frac{3}{2} \cdot 4$

x = 6

we see that the equation is equivalent to one of the form x = a . This is a vertical line, which has undefined slope.

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Example Question #1432 : Intermediate Geometry

What line is perpendicular to $y=\frac{-1}{2}x-3$ through (1,5) ?

Possible Answers:

2x+y=3

3x-y=4

x+3y=1

x-y=5

2x-y = -3

Correct answer:

2x-y = -3

Explanation:

$y=\frac{-1}{2}x-3$ is given in the slope-intercept form. So the slope is $\frac{-1}{2}$ and the y-intercept is .

If the lines are perpendicular, then $m_{1}\cdot m_{2}=-1$ so the new slope must be

Next we substitute the new slope and the given point into the slope-intercept form of the equation to calculate the intercept. So the equation to solve becomes 5=2(1) +b so b=3

So the equation of the perpendicular line becomes y=2x+3 or in standard form 2x-y=-3

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

Which line below is perpendicular to 5x+6y=18 ?

Possible Answers:

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

$y=-\frac{6}{5}x+8$

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

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

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

Correct answer:

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

Explanation:

The definition of a perpendicular line is one that has a negative, reciprocal slope to another.

For this particular problem, we must first manipulate our initial equation into a more easily recognizable and useful form: slope-intercept form or y=mx+b .

5x+6y=18

6y=-5x+18

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

According to our y=mx+b formula, our slope for the original line is $-\frac{5}{6}$ . We are looking for an answer that has a perpendicular slope, or an opposite reciprocal. The opposite reciprocal of $-\frac{5}{6}$ is $\frac{6}{5}$ . Flip the original and multiply it by .

Our answer will have a slope of $\frac{6}{5}$ . Search the answer choices for $\frac{6}{5}$ in the position of the y=mx+b equation.

$\dpi{100} y = \frac{6}{5}x + 3$ is our answer.

(As an aside, the negative reciprocal of 4 is $-\frac{1}{4}$ . Place the whole number over one and then flip/negate. This does not apply to the above problem, but should be understood to tackle certain permutations of this problem type where the original slope is an integer.)

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Example Question #1433 : Intermediate Geometry

Find the equation of the line perpendicular to $y=\frac{-1}{3}x+2$ .

Possible Answers:

y=3x-2

y=-3x-2

y=3x+2

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

y=-3x+2

Correct answer:

y=3x+2

Explanation:

The definition of a perpendicular line is a line with a negative reciprocal slope and identical intercept.

Therefore we need a line with slope 3 and intercept 2.

This means the only fitting line is y=3x+2 .

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Example Question #1434 : Intermediate Geometry

Which one of these equations is perpendicular to:

y=x+2

Possible Answers:

y=-x^2

y=-x+1

y=x

y=-2x+5

y=2

Correct answer:

y=-x+1

Explanation:

To find the perpendicular line to

y=x+2

we need to find the negative reciprocal of the slope of the above equation.

So the slope of the above equation is since changes by when is incremented.

The negative reciprocal is:

$reciprocal=\frac{-1}{slope}$

$reciprocal=\frac{1}{-1}$

reciprocal=-1

So we are looking for an equation with a .

Only y=-x+1 satisfies this condition.

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Example Question #1435 : Intermediate Geometry

Suppose a line is represented by a function f(x) = 2x+4 . Find the equation of a perpendicular line that intersects the point (1,2) .

Possible Answers:

$f(x)= -\frac{1}{2}x$

$f(x)= -\frac{1}{2}x+1\frac{1}{2}$

$f(x)= -\frac{1}{2}x+2\frac{1}{2}$

f(x)=2x

$f(x)= \frac{1}{2}x-1\frac{1}{2}$

Correct answer:

$f(x)= -\frac{1}{2}x+2\frac{1}{2}$

Explanation:

Determine the slope of the function f(x) = 2x+4 . The slope is: m1=2

The slope of a perpendicular line is the negative reciprocal of the original slope. Determine the value of the slope perpendicular to the original function.

$m2 = - \frac{1}{m1} = -\frac{1}{2}$

Plug in the given point and the slope to the slope-intercept form to find the y-intercept.

y=mx+b

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

$b=2\frac{1}{2}$

Substitute the slope of the perpendicular line and the new y-intercept back in the slope-intercept equation, y=mx+b .

The correct answer is: $f(x)= -\frac{1}{2}x+2\frac{1}{2}$

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Intermediate Geometry : Intermediate Geometry

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

Example Questions

Example Question #143 : Lines

Example Question #72 : Other Lines

Example Question #1431 : Intermediate Geometry

Example Question #141 : Coordinate Geometry

Example Question #141 : Coordinate Geometry

Example Question #1432 : Intermediate Geometry

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

Example Question #1433 : Intermediate Geometry

Example Question #1434 : Intermediate Geometry

Example Question #1435 : Intermediate Geometry

All Intermediate Geometry Resources

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