Lines - High School Math

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Question

Which of the following lines is parallel to the line $y=\frac{1}{4}x-3$ ?

Answer

Parallel lines have the same slope. In slope-intercept form, , is the slope.

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

Here the slope is $\frac{1}{4}$ ; thus, any line that is parallel to the line in question will also have a slope of $\frac{1}{4}$ .

Only one answer choice satisfies this requirement: $y= \frac {1}{4} x-20$

Note: the answer choice $y= \frac {1}{4}$ is incorrect. If put into form, the equation becomes $y =0x+\frac{1}{4}$ . Therefore the slope is actually , not $\frac{1}{4}$ .

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Question

Which of the following lines is parallel to ?

Answer

Two lines that are parallel have the same slope. The slope of is , so we want another line with a slope of . The only other line with a slope of is .

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Question

Which of these lines is parallel to $y=\frac{2}{3}x+\frac{5}{6}$ ?

Answer

Lines are parallel if they have the same slope. In standard form, is the slope.

For our given equation, the slope is $\frac{2}{3}$ . Only $y=\frac{2}{3}x-12$ has the same slope.

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Question

Which of these lines is parallel to $y=\frac{4}{7}x+\frac{1}{3}$ ?

Answer

Lines are parallel if they have the same slope. In standard form, is the slope.

For our given equation, the slope is $\frac{4}{7}$ . Only $y=\frac{4}{7}x-1$ has the same slope.

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Question

Which of the following lines will be parallel to $y=\frac{3}{4}x+12$ ?

Answer

Two lines are parallel if they have the same slope. When a line is in standard form, the is the slope.

For the given line $y=\frac{3}{4}x+12$ , the slope will be $\frac{3}{4}$ . Only one other line has a slope of $\frac{3}{4}$ : $y=\frac{3}{4}x-5$

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Question

Are the following lines parallel?

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

Answer

By definition, two lines are parallel if they have the same slope. Notice that since we are given the lines in the format, and our slope is given by , it is clear that the slopes are not the same in this case, and thus the lines are not parallel.

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Question

Which of the following are perpendicular to the line with the formula $y = -\frac{1}{3}X + 3$ ?

I.

II. $y = \frac{1}{3}X + 3$

III. $y = \frac{4.5}{1.5}X -3$

Answer

The slope of a perpendicular line is equal to the negative reciprocal of the original line. This means that the slope of our perpendicular line must be 3. We can also note that $\frac{4.5}{1.5}$ is also equal to 3, so both of these slopes are correct. The y-intercept does not matter, as the slope is the only thing that determines the slant of the line. Therefore, numerals I and III are both correct.

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Question

Which of the following lines is perpendicular to ?

Answer

In order for two lines to be perpendicular to each other, their slopes must be opposites and reciprocals of each other, meaning the fraction must be flipped upside down and the signs must be changed. In this situation, the original equation had a slope of , so the perpendicular slope must be $-\frac{1}{2}$ .

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Question

Which of the following lines will be perpendicular to ?

Answer

Two lines are perpendicular if they have opposite reciprocal slopes. When a line is in standard form, the is the slope. A perpendicular line will have a slope of $-\frac{1}{m}$ .

The slope of our given line is . Therefore we want a slope of $-\frac{1}{2}$ . The only line with the correct slope is $y=-\frac{1}{2}x+1$ .

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Question

The points A, B, and C reside on a line segment. B is the midpoint of AC. If line AB measures 6 units in length, what is the length of line AC?

Answer

If B is the midpoint of AC, then AC is twice as long as AB. We are told that AB=6.

The diagram shows six units between points A and B, with B as the midpoint of segment AC. Therefore segment BC is also six units long, so line AC is twelve units long.

$\small (2)(6)=12$

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Question

What is the length of a line with endpoints and ?

Answer

The formula for the length of a line is very similiar to the pythagorean theorem:

Plug in our given numbers to solve:

$\sqrt{169}=\sqrt{l^2}$

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Question

What line is parallel to through the point ?

Answer

The given line can be rewritten as $y=-\frac{1}{2}x+\frac{7}{10}$ , which has slope $m=-\frac{1}{2}$ .

If the new line is parallel to the old line, it must have the same slope. So we use the point-slope form of an equation to calculate the new intercept.

becomes $7=-\frac{1}{2}(-6)+b$ where .

So the equation of the parallel line is $y=-\frac{1}{2}x+4$ .

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Question

Find the equation of a line parallel to the line that goes through points $\left ( 2,4 \right )$ and $\left ( 0,7 \right )$ .

Answer

Parallel lines share the same slope. Because the slope of the original line is $-\frac{3}{2}$ , the correct answer must have that slope, so the correct answer is

$y=-\frac{3}{2}X+7$

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Question

Find the equation of a line parallel to .

Answer

Since parallel lines share the same slope, the only answer that works is

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Question

Given the equation and the point , find a line through the point that is parallel to the given line.

Answer

In order for two lines to be parallel, they must have the same slope. The slope of the given line is , so we know that the line going through the given point also has to have a slope of . Using the point-slope formula,

$y-y_{1}=m(x-x_1)$ ,

where represents the slope and and represent the given points, plug in the points given and simplify into standard form:

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Question

What line is parallel to through ?

Answer

Parallel lines have the same slopes. The slope for the given equation is . We can use the slope and the new point in the slope intercept equation to solve for the intercept:

Therefore the new equation becomes:

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Question

What line is parallel to $y=\frac{1}{2}x-2$ through ?

Answer

Parallel lines have the same slope. The slope of the given line is $\frac{1}{2}$ .

Find the line with slope $\frac{1}{2}$ through the point by plugging this informatuon into the slope intercept equation, :

$7=\frac{1}{2}(6)+b$ , which gives .

Solve for by subtracting from both sides to get .

Then the parallel line equation becomes $y=\frac{1}{2}x+4$ , and converting to standard form gives .

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Question

What is the equation, in slope-intercept form, of the perpendicular bisector of the line segment that connects the points $\small (2,2)$ and $\small (8,6)$ ?

Answer

First, calculate the slope of the line segment between the given points.

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

We want a line that is perpendicular to this segment and passes through its midpoint. The slope of a perpendicular line is the negative inverse. The slope of the perpendicular bisector will be $m=-\frac{3}{2}$ .

Next, we need to find the midpoint of the segment, using the midpoint formula.

$mid=(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{x})=(\frac{2+8}{2},\frac{2+6}{2})=(5,4)$

Using the midpoint and the slope, we can solve for the value of the y-intercept.

$4=(-\frac{3}{2})(5)+b$

$4=\frac{8}{2}=-\frac{15}{2}+b$

$\frac{23}{2}=b$

Using this value, we can write the equation for the perpendicular bisector in slope-intercept form.

$y=-\frac{3}{2}x+\frac{23}{2}$

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Question

Write an equation in slope-intercept form for the line that passes through $\left ( 3,1 \right )$ and that is perpendicular to a line which passes through the two points $\left ( 3,1 \right )$ and $\left ( -2,5 \right )$ .

Answer

Find the slope of the line through the two points. It is $-\frac{4}{5}$ .

Since the slope of a perpendicular line is the negative reciprocal of the original line, the new line's slope is $\frac{5}{4}$ . Plug the slope and one of the points into the point-slope formula $y-y_{1}=m(x-x_{1})$ . Isolate for .

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Question

Find the equation of a line perpendicular to

Answer

Since a perpendicular line has a slope that is the negative reciprocal of the original line, the new slope is $\frac{1}{3}$ . There is only one answer with the correct slope.

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