Geometry - Lines | Practice Hub

Suppose a line is represented by a function . Find the equation of a perpendicular line that intersects the point .

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

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

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

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

Explanation

Determine the slope of the function . The slope is:

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.

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

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

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

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

Transverselines

Which answer contains all the angles (other than itself) that are congruent to Angle 1?

Angles 4, 5, and 8

Angles 2 and 4

Angles 2 and 5

Angles 8 and 6

Angles 4 and 5

Explanation

Because of the Corresponding Angles Theorem (Angle 2 and Angle 5), Alternate Exterior Angles (Angle 2 and Angle 8), and Vertical Angles (Angle 2 and Angle 4).

Write the equation for the line passing through the points and

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

$y = -\frac{1}{2}x + 5$

$y = \frac{1}{2} x + 5$

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

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

Explanation

To determine the equation, first find the slope:

$\frac{\Delta y}{\Delta x} = \frac{3--1}{-4-4} = \frac{4 }{-8 } =- \frac{1}{2}$

We want this equation in slope-intercept form, . We know and because we have two coordinate pairs to choose from representing an and a . We know because that represents the slope. We just need to solve for , and then we can write the equation.

We can choose either point and get the correct answer. Let's choose :

$-1 = -\frac{1}{2} (4) + b$ multiply ""

add to both sides

This means that the form is $y = -\frac{1}{2}x +1$

If the -intercept of a line is , and the -intercept is , what is the equation of this line?

$y=-\frac{1}{5}x+5$

$y=\frac{1}{5}x+5$

Explanation

If the y-intercept of a line is , then the -value is when is zero. Write the point:

If the -intercept of a line is , then the -value is when is zero. Write the point:

Use the following formula for slope and the two points to determine the slope:

$m=\frac{y_2-y_1}{x_2-x_1} = \frac{0-5}{-1-0}= \frac{-5}{-1} =5$

Use the slope intercept form and one of the points, suppose , to find the equation of the line by substituting in the values of the point and solving for , the -intercept.

Therefore, the equation of this line is .

Transverselines

Which answer contains all the angles (other than itself) that are congruent to Angle 1?

Angles 4, 5, and 8

Angles 2 and 4

Angles 2 and 5

Angles 8 and 6

Angles 4 and 5

Explanation

Because of the Corresponding Angles Theorem (Angle 2 and Angle 5), Alternate Exterior Angles (Angle 2 and Angle 8), and Vertical Angles (Angle 2 and Angle 4).

If the -intercept of a line is , and the -intercept is , what is the equation of this line?

$y=-\frac{1}{5}x+5$

$y=\frac{1}{5}x+5$

Explanation

If the y-intercept of a line is , then the -value is when is zero. Write the point:

If the -intercept of a line is , then the -value is when is zero. Write the point:

Use the following formula for slope and the two points to determine the slope:

$m=\frac{y_2-y_1}{x_2-x_1} = \frac{0-5}{-1-0}= \frac{-5}{-1} =5$

Use the slope intercept form and one of the points, suppose , to find the equation of the line by substituting in the values of the point and solving for , the -intercept.

Therefore, the equation of this line is .

Write the equation for the line passing through the points and

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

$y = -\frac{1}{2}x + 5$

$y = \frac{1}{2} x + 5$

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

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

Explanation

To determine the equation, first find the slope:

$\frac{\Delta y}{\Delta x} = \frac{3--1}{-4-4} = \frac{4 }{-8 } =- \frac{1}{2}$

We can choose either point and get the correct answer. Let's choose :

$-1 = -\frac{1}{2} (4) + b$ multiply ""

add to both sides

This means that the form is $y = -\frac{1}{2}x +1$

What is the slope of the line that passes through the points and ?

Explanation

The slope of a line is sometimes referred to as "rise over run." This is because the formula for slope is the change in y-value (rise) divided by the change in x-value (run). Therefore, if you are given two points, $(x_{1},y_{1})$ and $(x_{2},y_{2})$ , the slope of their line can be found using the following formula: $slope=(y_{1}-y_{2})/(x_{1}-x_{2})$

This gives us .

Suppose a line is represented by a function . Find the equation of a perpendicular line that intersects the point .

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

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

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

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

Explanation

Determine the slope of the function . The slope is:

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.

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

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

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

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

Transverselines

Which answer contains all the angles (other than itself) that are congruent to Angle 1?

Angles 4, 5, and 8

Angles 2 and 4

Angles 2 and 5

Angles 8 and 6

Angles 4 and 5

Explanation

Because of the Corresponding Angles Theorem (Angle 2 and Angle 5), Alternate Exterior Angles (Angle 2 and Angle 8), and Vertical Angles (Angle 2 and Angle 4).

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