Question Find the value of $x$ such that the line segment $\overline{AB}$ is parallel to the vector $\overrightarrow{CD}$ , given points $A(-9,-12), B(-2,2), C(x, 6), D(-5,-2)$ .

Studdy Solution

STEP 1

Assumptions1. The coordinates of points A, B, and D are given as A(-9,-12), B(-,), and D(-5,-) respectively. . The coordinates of point C are given as C(x,6), where x is unknown.
3. The line segment AB is parallel to the vector CD.

STEP 2

For two lines to be parallel, their slopes must be equal. Therefore, the slope of AB must be equal to the slope of CD.
The formula for the slope of a line given two points (x1, y1) and (x2, y2) is $lope = \frac{y2 - y1}{x2 - x1}$

STEP 3

Let's first calculate the slope of AB using the coordinates of points A and B.
$lope_{AB} = \frac{y_B - y_A}{x_B - x_A}$

STEP 4

Substitute the coordinates of points A and B into the formula.
$lope_{AB} = \frac{2 - (-12)}{-2 - (-9)}$

STEP 5

implify the equation to find the slope of AB.
$lope_{AB} = \frac{2 +12}{-2 +9} = \frac{14}{7} =2$

STEP 6

Now, let's calculate the slope of CD using the coordinates of points C and D.
$lope_{CD} = \frac{y_D - y_C}{x_D - x_C}$

STEP 7

Substitute the coordinates of points C and D into the formula.
$lope_{CD} = \frac{-2 -6}{-5 - x}$

STEP 8

implify the equation to find the slope of CD.
$lope_{CD} = \frac{-8}{-5 - x}$

STEP 9

Since AB is parallel to CD, the slopes must be equal. Therefore, we can set the two slopes equal to each other and solve for x.
$lope_{AB} = Slope_{CD}$ $2 = \frac{-8}{-5 - x}$

STEP 10

Cross multiply to solve for x.
$2(-5 - x) = -8$

STEP 11

implify the equation to solve for x.
$-10 -x = -8$

STEP 12

Add10 to both sides of the equation.
$-2x = -8 +10$

STEP 13

implify the equation to solve for x.
$-2x =2$

STEP 14

Divide both sides of the equation by -2 to solve for x.
$x = \frac{2}{-2}$

STEP 15

implify to find the value of x.
$x = -$ The value of x that makes AB parallel to CD is -.

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Question Find the value of $x$ such that the line segment $\overline{AB}$ is parallel to the vector $\overrightarrow{CD}$ , given points $A(-9,-12), B(-2,2), C(x, 6), D(-5,-2)$ .

Studdy Solution

STEP 1

Assumptions1. The coordinates of points A, B, and D are given as A(-9,-12), B(-,), and D(-5,-) respectively. . The coordinates of point C are given as C(x,6), where x is unknown.
3. The line segment AB is parallel to the vector CD.

STEP 2

For two lines to be parallel, their slopes must be equal. Therefore, the slope of AB must be equal to the slope of CD.
The formula for the slope of a line given two points (x1, y1) and (x2, y2) is $lope = \frac{y2 - y1}{x2 - x1}$

STEP 3

Let's first calculate the slope of AB using the coordinates of points A and B.
$lope_{AB} = \frac{y_B - y_A}{x_B - x_A}$

STEP 4

Substitute the coordinates of points A and B into the formula.
$lope_{AB} = \frac{2 - (-12)}{-2 - (-9)}$

STEP 5

implify the equation to find the slope of AB.
$lope_{AB} = \frac{2 +12}{-2 +9} = \frac{14}{7} =2$

STEP 6

Now, let's calculate the slope of CD using the coordinates of points C and D.
$lope_{CD} = \frac{y_D - y_C}{x_D - x_C}$

STEP 7

Substitute the coordinates of points C and D into the formula.
$lope_{CD} = \frac{-2 -6}{-5 - x}$

STEP 8

implify the equation to find the slope of CD.
$lope_{CD} = \frac{-8}{-5 - x}$

STEP 9

Since AB is parallel to CD, the slopes must be equal. Therefore, we can set the two slopes equal to each other and solve for x.
$lope_{AB} = Slope_{CD}$ $2 = \frac{-8}{-5 - x}$

STEP 10

Cross multiply to solve for x.
$2(-5 - x) = -8$

STEP 11

implify the equation to solve for x.
$-10 -x = -8$

STEP 12

Add10 to both sides of the equation.
$-2x = -8 +10$

STEP 13

implify the equation to solve for x.
$-2x =2$

STEP 14

Divide both sides of the equation by -2 to solve for x.
$x = \frac{2}{-2}$

STEP 15

implify to find the value of x.
$x = -$ The value of x that makes AB parallel to CD is -.

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Question Find the value of xxx such that the line segment AB‾\overline{AB}AB is parallel to the vector CDundefined\overrightarrow{CD}CD, given points A(−9,−12),B(−2,2),C(x,6),D(−5,−2)A(-9,-12), B(-2,2), C(x, 6), D(-5,-2)A(−9,−12),B(−2,2),C(x,6),D(−5,−2).

Studdy Solution

STEP 1

Assumptions1. The coordinates of points A, B, and D are given as A(-9,-12), B(-,), and D(-5,-) respectively. . The coordinates of point C are given as C(x,6), where x is unknown.3. The line segment AB is parallel to the vector CD.

STEP 2

For two lines to be parallel, their slopes must be equal. Therefore, the slope of AB must be equal to the slope of CD.The formula for the slope of a line given two points (x1, y1) and (x2, y2) islope=y2−y1x2−x1lope = \frac{y2 - y1}{x2 - x1}lope=x2−x1y2−y1​

STEP 3

Let's first calculate the slope of AB using the coordinates of points A and B.lopeAB=yB−yAxB−xAlope_{AB} = \frac{y_B - y_A}{x_B - x_A}lopeAB​=xB​−xA​yB​−yA​​

STEP 4

Substitute the coordinates of points A and B into the formula.lopeAB=2−(−12)−2−(−9)lope_{AB} = \frac{2 - (-12)}{-2 - (-9)}lopeAB​=−2−(−9)2−(−12)​

STEP 5

implify the equation to find the slope of AB.lopeAB=2+12−2+9=147=2lope_{AB} = \frac{2 +12}{-2 +9} = \frac{14}{7} =2lopeAB​=−2+92+12​=714​=2

STEP 6

Now, let's calculate the slope of CD using the coordinates of points C and D.lopeCD=yD−yCxD−xClope_{CD} = \frac{y_D - y_C}{x_D - x_C}lopeCD​=xD​−xC​yD​−yC​​

STEP 7

Substitute the coordinates of points C and D into the formula.lopeCD=−2−6−5−xlope_{CD} = \frac{-2 -6}{-5 - x}lopeCD​=−5−x−2−6​

STEP 8

implify the equation to find the slope of CD.lopeCD=−8−5−xlope_{CD} = \frac{-8}{-5 - x}lopeCD​=−5−x−8​

STEP 9

Since AB is parallel to CD, the slopes must be equal. Therefore, we can set the two slopes equal to each other and solve for x.lopeAB=SlopeCDlope_{AB} = Slope_{CD}lopeAB​=SlopeCD​2=−8−5−x2 = \frac{-8}{-5 - x}2=−5−x−8​

STEP 10

Cross multiply to solve for x.2(−5−x)=−82(-5 - x) = -82(−5−x)=−8

STEP 11

implify the equation to solve for x.−10−x=−8-10 -x = -8−10−x=−8

STEP 12

Add10 to both sides of the equation.−2x=−8+10-2x = -8 +10−2x=−8+10

STEP 13

implify the equation to solve for x.−2x=2-2x =2−2x=2

STEP 14

Divide both sides of the equation by -2 to solve for x.x=2−2x = \frac{2}{-2}x=−22​

STEP 15

implify to find the value of x.x=−x = -x=−The value of x that makes AB parallel to CD is -.

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Question Find the value of $x$ such that the line segment $\overline{AB}$ is parallel to the vector $\overrightarrow{CD}$ , given points $A(-9,-12), B(-2,2), C(x, 6), D(-5,-2)$ .

Assumptions1. The coordinates of points A, B, and D are given as A(-9,-12), B(-,), and D(-5,-) respectively. . The coordinates of point C are given as C(x,6), where x is unknown.
3. The line segment AB is parallel to the vector CD.

For two lines to be parallel, their slopes must be equal. Therefore, the slope of AB must be equal to the slope of CD.
The formula for the slope of a line given two points (x1, y1) and (x2, y2) is $lope = \frac{y2 - y1}{x2 - x1}$

Let's first calculate the slope of AB using the coordinates of points A and B.
$lope_{AB} = \frac{y_B - y_A}{x_B - x_A}$

Substitute the coordinates of points A and B into the formula.
$lope_{AB} = \frac{2 - (-12)}{-2 - (-9)}$

implify the equation to find the slope of AB.
$lope_{AB} = \frac{2 +12}{-2 +9} = \frac{14}{7} =2$

Now, let's calculate the slope of CD using the coordinates of points C and D.
$lope_{CD} = \frac{y_D - y_C}{x_D - x_C}$

Substitute the coordinates of points C and D into the formula.
$lope_{CD} = \frac{-2 -6}{-5 - x}$

implify the equation to find the slope of CD.
$lope_{CD} = \frac{-8}{-5 - x}$

Since AB is parallel to CD, the slopes must be equal. Therefore, we can set the two slopes equal to each other and solve for x.
$lope_{AB} = Slope_{CD}$ $2 = \frac{-8}{-5 - x}$

Cross multiply to solve for x.
$2(-5 - x) = -8$

implify the equation to solve for x.
$-10 -x = -8$

Add10 to both sides of the equation.
$-2x = -8 +10$

implify the equation to solve for x.
$-2x =2$

Divide both sides of the equation by -2 to solve for x.
$x = \frac{2}{-2}$

implify to find the value of x.
$x = -$ The value of x that makes AB parallel to CD is -.