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Intermediate Geometry : Intermediate Geometry

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Example Questions

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

Find the equation of the line parallel to 3x+3y=1 that passes through the point (2,2) .

Possible Answers:

y=-x+4

y=x-2

y=3x+4

y=3x-4

y=-x-4

Correct answer:

y=-x+4

Explanation:

Write 3x+3y=1 in slope intercept form, y=mx+b , to determine the slope, :

3y=-3x+1

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

The slope is:

m=-1

Given the slope, use the point (2,2) and the equation y=mx+b to solve for the value of the -intercept, . Substitute the known values.

2=(-1)(2)+b

b=4

With the known slope and the -intercept, plug both values back to the slope intercept formula. The answer is y=-x+4 .

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Example Question #2 : How To Find The Equation Of A Parallel Line

Given y=3x+20 , find the equation of a line parallel.

Possible Answers:

$y=\frac{-1}{3}-20$

y=-3x-12

$y=\frac{1}{3}x-20$

y=3x-12

y=0.3x+20

Correct answer:

y=3x-12

Explanation:

The definition of a parallel line is that the lines have the same slopes, but different intercepts. The only answer with the same slope is y=3x-12 .

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

Which one of these equations is parallel to:

y=3x+4

Possible Answers:

y=-3x+7

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

y=4

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

y=3x+1

Correct answer:

y=3x+1

Explanation:

Equations that are parallel have the same slope.

For the equation:

y=3x+4

The slope is since that is how much changes with increment of .

The only other equation with a slope of is:

y=3x+1

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Example Question #2 : How To Find The Equation Of A Parallel Line

What equation is parallel to:

y=x

Possible Answers:

y=x+42746

y=-x+1

y=2x

y=x^2+4

y=0

Correct answer:

y=x+42746

Explanation:

To find a parallel line to

y=x

we need to find another equation with the same slope of or .

The only equation that satisfies this is y=x+42746 .

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Example Question #3 : How To Find The Equation Of A Parallel Line

What equation is parallel to:

y=5x+5

Possible Answers:

y=5x+10

$y=\frac{1}{5}x+4$

y=4x+3

y=x+5

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

Correct answer:

y=5x+10

Explanation:

To find an equation that is parallel to

y=5x+5

we need to find an equation with the same slope of .

Basically we are looking for another equation with .

The only other equation that satisfies this is

y=5x+10 .

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Example Question #7 : How To Find The Equation Of A Parallel Line

A line is parallel to the line of the equation

5x+ 7y = 32

and passes through the point (-2, 8) .

Give the equation of the line in standard form.

Possible Answers:

7x- 5y= -54

5x+ 7y = 46

5x- 7y = -66

None of the other choices gives the correct response.

7x + 5y= 26

Correct answer:

5x+ 7y = 46

Explanation:

Two parallel lines have the same slope. Therefore, it is necessary to find the slope of the line of the equation

5x+ 7y = 32

Rewrite the equation in slope-intercept form y = mx+ b . , the coefficient of , will be the slope of the line.

Add -5x to both sides:

5x+ 7y + (-5x ) = 32 + (-5x )

7y = -5x+32

Multiply both sides by $\frac{1}{7}$ , distributing on the right:

$\frac{1}{7} \cdot 7y = \frac{1}{7} \cdot( -5x+32)$

$y = -\frac{5}{7}x+\frac{32}{7}$

The slope of this line is $-\frac{5}{7}$ . The slope of the first line will be the same. The slope-intercept form of the equation of this line will be

$y =-\frac{5}{7}x+b$ .

To find , set x = -2 and y = 8 and solve:

$8 =-\frac{5}{7} (-2)+b$

$8 = \frac{10}{7} +b$

$\frac{56}{7} = \frac{10}{7} +b$

Subtract $\frac{10}{7}$ from both sides:

$\frac{56}{7} - \frac{10}{7} = \frac{10}{7} +b - \frac{10}{7}$

$b = \frac{46}{7}$

The slope-intercept form of the equation is $y =-\frac{5}{7}x+ \frac{46}{7}$

To rewrite in standard form with integer coefficients:

Multiply both sides by 7:

$7 y =7\left (-\frac{5}{7}x+ \frac{46}{7} \right )$

$7 y =7\left (-\frac{5}{7}x \right ) +7\left ( \frac{46}{7} \right )$

7 y =-5x +46

Add to both sides:

7 y + 5x =-5x +46 + 5x

5x+ 7y = 46 ,

the correct equation in standard form.

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

Given the function y=3x+10 , find the equation of the tangent line passing through x=2 .

Possible Answers:

y=3x+6

y=3x

y=3x+10

$\textup{The equation does not exist.}$

y=3x+16

Correct answer:

y=3x+10

Explanation:

Find the slope of y=3x+10 . The slope is 3.

Substitute x=2 to determine the y-value.

y=3x+10=3(2)+10=16

The point is (2,16) .

Use the slope-intercept formula to find the y-intercept, given the point and slope.

y=mx+b

Substitute the point and the slope.

16=3(2)+b

b=10

Substitute the y-intercept and slope back to the slope-intercept formula.

The correct answer is: y=3x+10

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

Find the equation of the tangent line at the point (2,4) if the given function is 2y=3x+4 .

Possible Answers:

$y=\frac{3}{2}x+4$

y=3x+1

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

y=3x+2

y=3x-2

Correct answer:

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

Explanation:

Write 2y=3x+4 in slope-intercept form y=mx+b and determine the slope.

Rearranging our equation to be in slope-intercept form we get:

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

The slope is our value which is $\frac{3}{2}$ .

Substitute the slope and the point to the slope-intercept form.

y=mx+b

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

b=1

Substitute the slope and y-intercept to find our final equation.

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

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

What is the equation of the tangent line at x=0 to the equation 5y-3x+2=0 ?

Possible Answers:

y=-3x-2

$y=\frac{3}{5}x-1$

y=-3x-1

y=-3x+1

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

Correct answer:

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

Explanation:

Rewrite 5y-3x+2=0 in slope-intercept form to determine the slope. Remember slope-intercept form is y=mx+b .

Therefore, the equation becomes,

5y=3x-2

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

The slope is the value of the function thus, it is $\frac{3}{5}$ .

Substitute x=0 back to the original equation to find the value of .

5y-3(0)+2=0

5y+2=0

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

Substitute the point $\left(0,-\frac{2}{5}\right)$ and the slope of the line into the slope-intercept equation.

y=mx+b

$-\frac{2}{5}=\left(\frac{3}{5}\right)(0)+b$

$b=-\frac{2}{5}$

Substitute the point and the slope back in to the slope-intercept formula.

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

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Example Question #2 : How To Find The Equation Of A Tangent Line

Write the equation for the tangent line of the circle x^2 + (y-3)^2 = 41 through the point (4, -2) .

Possible Answers:

y = 4x - 18

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

$y = \frac{4}{5} x + 3$

$y = \frac{4}{5}x - \frac{26}{5 }$

y = 4x + 3

Correct answer:

$y = \frac{4}{5}x - \frac{26}{5 }$

Explanation:

The circle's center is (0,3) . The tangent line will be perpendicular to the line going through the points (0,3) and (4, -2) , so it will be helpful to know the slope of this line:

$\frac{ \Delta y }{ \Delta x } = \frac{ 3 - - 2 }{ 0 - 4 } = \frac{ 5 }{ -4 }$

Since the tangent line is perpendicular, its slope is $\frac{4}{5 }$

To write the equation in the form y = mx + b , we need to solve for "b," the y-intercept. We can plug in the slope for "m" and the coordinates of the point (4, -2) for x and y:

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

$-2 = \frac{16} { 5 } + b$

$-\frac{10}{5} = \frac{16} {5} + b$

$-\frac{26}{5} = b$

The equation is $y = \frac{4}{5} x - \frac{26} {5 }$

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Intermediate Geometry : Intermediate Geometry

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

Example Questions

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

Example Question #2 : How To Find The Equation Of A Parallel Line

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

Example Question #2 : How To Find The Equation Of A Parallel Line

Example Question #3 : How To Find The Equation Of A Parallel Line

Example Question #7 : How To Find The Equation Of A Parallel Line

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

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

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

Example Question #2 : How To Find The Equation Of A Tangent Line

All Intermediate Geometry Resources

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