QuestionFind $A C$ if line $y$ bisects $A C$ , where $A B=4-5 x$ and $B C=2 x+25$ . Solve: $4-5 x=2 x+25$

Studdy Solution

STEP 1

Assumptions1. Line $y$ bisects $AC$ , which means $AB = BC$ . . The length of $AB$ is given by $4-5x$ .
3. The length of $BC$ is given by $x+25$ .

STEP 2

Since $AB = BC$ , we can set the expressions for $AB$ and $BC$ equal to each other and solve for $x$ .
$4-5x =2x+25$

STEP 3

To isolate $x$ , we can start by adding $5x$ to both sides of the equation to get rid of the negative $5x$ on the left side.
$=7x +25$

STEP 4

Next, subtract $25$ from both sides to isolate $7x$ on one side of the equation.
$-21 =7x$

STEP 5

Finally, divide both sides by $7$ to solve for $x$ .
$x = -3$

STEP 6

Now that we have the value of $x$ , we can substitute it into the expressions for $AB$ and $BC$ to find the lengths of these segments.
For $AB$ , we have $AB =4 -5(-3)$

STEP 7

Calculate the length of $AB$ .
$AB =4 +15 =19$

STEP 8

For $BC$ , we have $BC =2(-3) +25$

STEP 9

Calculate the length of $BC$ .
$BC = -6 +25 =19$

STEP 10

Since $AB = BC$ , and line $y$ bisects $AC$ , the length of $AC$ is simply $2 \times AB$ or $2 \times BC$ .
$AC =2 \times AB$

STEP 11

Substitute the value of $AB$ into the equation to find $AC$ .
$AC = \times19$

STEP 12

Calculate the length of $AC$ .
$AC =2 \times19 =38$ So, the length of $AC$ is $38$ .

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QuestionFind $A C$ if line $y$ bisects $A C$ , where $A B=4-5 x$ and $B C=2 x+25$ . Solve: $4-5 x=2 x+25$

Studdy Solution

STEP 1

Assumptions1. Line $y$ bisects $AC$ , which means $AB = BC$ . . The length of $AB$ is given by $4-5x$ .
3. The length of $BC$ is given by $x+25$ .

STEP 2

Since $AB = BC$ , we can set the expressions for $AB$ and $BC$ equal to each other and solve for $x$ .
$4-5x =2x+25$

STEP 3

To isolate $x$ , we can start by adding $5x$ to both sides of the equation to get rid of the negative $5x$ on the left side.
$=7x +25$

STEP 4

Next, subtract $25$ from both sides to isolate $7x$ on one side of the equation.
$-21 =7x$

STEP 5

Finally, divide both sides by $7$ to solve for $x$ .
$x = -3$

STEP 6

Now that we have the value of $x$ , we can substitute it into the expressions for $AB$ and $BC$ to find the lengths of these segments.
For $AB$ , we have $AB =4 -5(-3)$

STEP 7

Calculate the length of $AB$ .
$AB =4 +15 =19$

STEP 8

For $BC$ , we have $BC =2(-3) +25$

STEP 9

Calculate the length of $BC$ .
$BC = -6 +25 =19$

STEP 10

Since $AB = BC$ , and line $y$ bisects $AC$ , the length of $AC$ is simply $2 \times AB$ or $2 \times BC$ .
$AC =2 \times AB$

STEP 11

Substitute the value of $AB$ into the equation to find $AC$ .
$AC = \times19$

STEP 12

Calculate the length of $AC$ .
$AC =2 \times19 =38$ So, the length of $AC$ is $38$ .

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QuestionFind ACA CAC if line yyy bisects ACA CAC, where AB=4−5xA B=4-5 xAB=4−5x and BC=2x+25B C=2 x+25BC=2x+25. Solve: 4−5x=2x+254-5 x=2 x+254−5x=2x+25

Studdy Solution

STEP 1

Assumptions1. Line yyy bisects ACACAC, which means AB=BCAB = BCAB=BC. . The length of ABABAB is given by 4−5x4-5x4−5x.3. The length of BCBCBC is given by x+25x+25x+25.

STEP 2

Since AB=BCAB = BCAB=BC, we can set the expressions for ABABAB and BCBCBC equal to each other and solve for xxx.4−5x=2x+254-5x =2x+254−5x=2x+25

STEP 3

To isolate xxx, we can start by adding 5x5x5x to both sides of the equation to get rid of the negative 5x5x5x on the left side.=7x+25 =7x +25=7x+25

STEP 4

Next, subtract 252525 from both sides to isolate 7x7x7x on one side of the equation.−21=7x-21 =7x−21=7x

STEP 5

Finally, divide both sides by 777 to solve for xxx.x=−3x = -3x=−3

STEP 6

Now that we have the value of xxx, we can substitute it into the expressions for ABABAB and BCBCBC to find the lengths of these segments.For ABABAB, we haveAB=4−5(−3)AB =4 -5(-3)AB=4−5(−3)

STEP 7

Calculate the length of ABABAB.AB=4+15=19AB =4 +15 =19AB=4+15=19

STEP 8

For BCBCBC, we haveBC=2(−3)+25BC =2(-3) +25BC=2(−3)+25

STEP 9

Calculate the length of BCBCBC.BC=−6+25=19BC = -6 +25 =19BC=−6+25=19

STEP 10

Since AB=BCAB = BCAB=BC, and line yyy bisects ACACAC, the length of ACACAC is simply 2×AB2 \times AB2×AB or 2×BC2 \times BC2×BC.AC=2×ABAC =2 \times ABAC=2×AB

STEP 11

Substitute the value of ABABAB into the equation to find ACACAC.AC=×19AC = \times19AC=×19

STEP 12

Calculate the length of ACACAC.AC=2×19=38AC =2 \times19 =38AC=2×19=38So, the length of ACACAC is 383838.

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QuestionFind $A C$ if line $y$ bisects $A C$ , where $A B=4-5 x$ and $B C=2 x+25$ . Solve: $4-5 x=2 x+25$

Assumptions1. Line $y$ bisects $AC$ , which means $AB = BC$ . . The length of $AB$ is given by $4-5x$ .
3. The length of $BC$ is given by $x+25$ .

Since $AB = BC$ , we can set the expressions for $AB$ and $BC$ equal to each other and solve for $x$ .
$4-5x =2x+25$

To isolate $x$ , we can start by adding $5x$ to both sides of the equation to get rid of the negative $5x$ on the left side.
$=7x +25$

Next, subtract $25$ from both sides to isolate $7x$ on one side of the equation.
$-21 =7x$

Finally, divide both sides by $7$ to solve for $x$ .
$x = -3$

Now that we have the value of $x$ , we can substitute it into the expressions for $AB$ and $BC$ to find the lengths of these segments.
For $AB$ , we have $AB =4 -5(-3)$

Calculate the length of $AB$ .
$AB =4 +15 =19$

For $BC$ , we have $BC =2(-3) +25$

Calculate the length of $BC$ .
$BC = -6 +25 =19$

Since $AB = BC$ , and line $y$ bisects $AC$ , the length of $AC$ is simply $2 \times AB$ or $2 \times BC$ .
$AC =2 \times AB$

Substitute the value of $AB$ into the equation to find $AC$ .
$AC = \times19$

Calculate the length of $AC$ .
$AC =2 \times19 =38$ So, the length of $AC$ is $38$ .