QuestionTriangles $\triangle ABC$ and $\triangle XYZ$ are similar with a ratio of $8:3$ . If $XY=9$ , find $YZ$ .

Studdy Solution

STEP 1

Assumptions1. Triangle ABC is similar to triangle XYZ. The ratio of similitude between the two triangles is833. The length of side XZ is1 less than the length of side AB4. The length of side BC is4 more than the length of side AC5. The length of side XY is9

STEP 2

Since the triangles are similar, their corresponding sides are in proportion. Therefore, we can set up the following proportion $\frac{AB}{XY} = \frac{BC}{YZ} = \frac{AC}{XZ} = \frac{8}{}$

STEP 3

We know that $XY =9$ , so we can substitute this value into the proportion to find $AB$ .
$AB = XY \times \frac{8}{3}$

STEP 4

Substitute the value of $XY$ into the equation to find $AB$ .
$AB =9 \times \frac{8}{3}$

STEP 5

Calculate the value of $AB$ .
$AB =9 \times \frac{8}{3} =24$

STEP 6

We know that $XZ = AB -1$ , so we can substitute the value of $AB$ into this equation to find $XZ$ .
$XZ = AB -1$

STEP 7

Substitute the value of $AB$ into the equation to find $XZ$ .
$XZ =24 -1$

STEP 8

Calculate the value of $XZ$ .
$XZ =24 -1 =23$

STEP 9

We know that $BC = AC +4$ , and we also know that $AC = XZ$ , so we can substitute these values into the equation to find $BC$ .
$BC = XZ +4$

STEP 10

Substitute the value of $XZ$ into the equation to find $BC$ .
$BC =23 +4$

STEP 11

Calculate the value of $BC$ .
$BC =23 +4 =27$

STEP 12

Now we can use the proportion to find $YZ$ . $YZ = \frac{BC}{AB} \times XY$

STEP 13

Substitute the values of $BC$ , $AB$ , and $XY$ into the equation to find $YZ$ .
$YZ = \frac{27}{24} \times9$

STEP 14

Calculate the value of $YZ$ .
$YZ = \frac{27}{24} \times9 =10.125$ The length of $YZ$ is10.125, which is not an option in the given choices. Therefore, the answer is. None of the above.

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QuestionTriangles $\triangle ABC$ and $\triangle XYZ$ are similar with a ratio of $8:3$ . If $XY=9$ , find $YZ$ .

Studdy Solution

STEP 1

Assumptions1. Triangle ABC is similar to triangle XYZ. The ratio of similitude between the two triangles is833. The length of side XZ is1 less than the length of side AB4. The length of side BC is4 more than the length of side AC5. The length of side XY is9

STEP 2

Since the triangles are similar, their corresponding sides are in proportion. Therefore, we can set up the following proportion $\frac{AB}{XY} = \frac{BC}{YZ} = \frac{AC}{XZ} = \frac{8}{}$

STEP 3

We know that $XY =9$ , so we can substitute this value into the proportion to find $AB$ .
$AB = XY \times \frac{8}{3}$

STEP 4

Substitute the value of $XY$ into the equation to find $AB$ .
$AB =9 \times \frac{8}{3}$

STEP 5

Calculate the value of $AB$ .
$AB =9 \times \frac{8}{3} =24$

STEP 6

We know that $XZ = AB -1$ , so we can substitute the value of $AB$ into this equation to find $XZ$ .
$XZ = AB -1$

STEP 7

Substitute the value of $AB$ into the equation to find $XZ$ .
$XZ =24 -1$

STEP 8

Calculate the value of $XZ$ .
$XZ =24 -1 =23$

STEP 9

We know that $BC = AC +4$ , and we also know that $AC = XZ$ , so we can substitute these values into the equation to find $BC$ .
$BC = XZ +4$

STEP 10

Substitute the value of $XZ$ into the equation to find $BC$ .
$BC =23 +4$

STEP 11

Calculate the value of $BC$ .
$BC =23 +4 =27$

STEP 12

Now we can use the proportion to find $YZ$ . $YZ = \frac{BC}{AB} \times XY$

STEP 13

Substitute the values of $BC$ , $AB$ , and $XY$ into the equation to find $YZ$ .
$YZ = \frac{27}{24} \times9$

STEP 14

Calculate the value of $YZ$ .
$YZ = \frac{27}{24} \times9 =10.125$ The length of $YZ$ is10.125, which is not an option in the given choices. Therefore, the answer is. None of the above.

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QuestionTriangles △ABC\triangle ABC△ABC and △XYZ\triangle XYZ△XYZ are similar with a ratio of 8:38:38:3. If XY=9XY=9XY=9, find YZYZYZ.

Studdy Solution

STEP 1

Assumptions1. Triangle ABC is similar to triangle XYZ. The ratio of similitude between the two triangles is833. The length of side XZ is1 less than the length of side AB4. The length of side BC is4 more than the length of side AC5. The length of side XY is9

STEP 2

Since the triangles are similar, their corresponding sides are in proportion. Therefore, we can set up the following proportionABXY=BCYZ=ACXZ=8\frac{AB}{XY} = \frac{BC}{YZ} = \frac{AC}{XZ} = \frac{8}{}XYAB​=YZBC​=XZAC​=8​

STEP 3

We know that XY=9XY =9XY=9, so we can substitute this value into the proportion to find ABABAB.AB=XY×83AB = XY \times \frac{8}{3}AB=XY×38​

STEP 4

Substitute the value of XYXYXY into the equation to find ABABAB.AB=9×83AB =9 \times \frac{8}{3}AB=9×38​

STEP 5

Calculate the value of ABABAB.AB=9×83=24AB =9 \times \frac{8}{3} =24AB=9×38​=24

STEP 6

We know that XZ=AB−1XZ = AB -1XZ=AB−1, so we can substitute the value of ABABAB into this equation to find XZXZXZ.XZ=AB−1XZ = AB -1XZ=AB−1

STEP 7

Substitute the value of ABABAB into the equation to find XZXZXZ.XZ=24−1XZ =24 -1XZ=24−1

STEP 8

Calculate the value of XZXZXZ.XZ=24−1=23XZ =24 -1 =23XZ=24−1=23

STEP 9

We know that BC=AC+4BC = AC +4BC=AC+4, and we also know that AC=XZAC = XZAC=XZ, so we can substitute these values into the equation to find BCBCBC.BC=XZ+4BC = XZ +4BC=XZ+4

STEP 10

Substitute the value of XZXZXZ into the equation to find BCBCBC.BC=23+4BC =23 +4BC=23+4

STEP 11

Calculate the value of BCBCBC.BC=23+4=27BC =23 +4 =27BC=23+4=27

STEP 12

Now we can use the proportion to find YZYZYZ.YZ=BCAB×XYYZ = \frac{BC}{AB} \times XYYZ=ABBC​×XY

STEP 13

Substitute the values of BCBCBC, ABABAB, and XYXYXY into the equation to find YZYZYZ.YZ=2724×9YZ = \frac{27}{24} \times9YZ=2427​×9

STEP 14

Calculate the value of YZYZYZ.YZ=2724×9=10.125YZ = \frac{27}{24} \times9 =10.125YZ=2427​×9=10.125The length of YZYZYZ is10.125, which is not an option in the given choices. Therefore, the answer is. None of the above.

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QuestionTriangles $\triangle ABC$ and $\triangle XYZ$ are similar with a ratio of $8:3$ . If $XY=9$ , find $YZ$ .

Since the triangles are similar, their corresponding sides are in proportion. Therefore, we can set up the following proportion $\frac{AB}{XY} = \frac{BC}{YZ} = \frac{AC}{XZ} = \frac{8}{}$

We know that $XY =9$ , so we can substitute this value into the proportion to find $AB$ .
$AB = XY \times \frac{8}{3}$

Substitute the value of $XY$ into the equation to find $AB$ .
$AB =9 \times \frac{8}{3}$

Calculate the value of $AB$ .
$AB =9 \times \frac{8}{3} =24$

We know that $XZ = AB -1$ , so we can substitute the value of $AB$ into this equation to find $XZ$ .
$XZ = AB -1$

Substitute the value of $AB$ into the equation to find $XZ$ .
$XZ =24 -1$

Calculate the value of $XZ$ .
$XZ =24 -1 =23$

We know that $BC = AC +4$ , and we also know that $AC = XZ$ , so we can substitute these values into the equation to find $BC$ .
$BC = XZ +4$

Substitute the value of $XZ$ into the equation to find $BC$ .
$BC =23 +4$

Calculate the value of $BC$ .
$BC =23 +4 =27$

Now we can use the proportion to find $YZ$ . $YZ = \frac{BC}{AB} \times XY$

Substitute the values of $BC$ , $AB$ , and $XY$ into the equation to find $YZ$ .
$YZ = \frac{27}{24} \times9$

Calculate the value of $YZ$ .
$YZ = \frac{27}{24} \times9 =10.125$ The length of $YZ$ is10.125, which is not an option in the given choices. Therefore, the answer is. None of the above.