Solving Similar Triangles

Question 1 of 5

The following two triangles are similar.

Solve for the missing sides

x

$x$ and

y

$y$

$x =$ $x=$ (6) $y =$ $y=$ (4)

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Mute

First, identify corresponding sides and angles of the two triangles.

Write out the proportions of corresponding sides to solve for $x$ $x$

$\frac{\color{#e85e00}{QP}}{\color{#e85e00}{MN}}=\frac{\color\green{x}}{\color\green{ON}}$			Definition of Similarity

$\frac{\color{#e85e00}{15}}{\color{#e85e00}{5}}=\frac{\color\green{x}}{\color\green{2}}$			Plug in the side lengths

$2 \times 15$ $2 xx 15$	$=$ $=$	$5 \times x$ $5 xx x$	Cross Multiply
$30$ $30$ $\div 5$ $divide 5$	$=$ $=$	$5 x$ $5x$ $\div 5$ $divide 5$	Divide both sides by $5$ $5$
$6$ $6$	$=$ $=$	$x$ $x$
$x$ $x$	$=$ $=$	$6$ $6$

Do the same to solve for $y$ $y$

$\frac{\color{#e85e00}{QP}}{\color{#e85e00}{MN}}=\frac{\color{#004ec4}{RP}}{\color{#004ec4}{y}}$			Definition of Similarity

$\frac{\color{#e85e00}{15}}{\color{#e85e00}{5}}=\frac{\color{#004ec4}{12}}{\color{#004ec4}{y}}$			Plug in the side lengths

$y \times 15$ $y xx 15$	$=$ $=$	$5 \times 12$ $5 xx 12$	Cross Multiply
$15 y$ $15y$	$=$ $=$	$60$ $60$
$15 y$ $15y$ $\div 15$ $divide 15$	$=$ $=$	$60$ $60$ $\div 15$ $divide 15$	Divide both sides by $15$ $15$
$y$ $y$	$=$ $=$	$4$

$x=6$ and

$y=4$

Question 2 of 5

The following two triangles are similar.

Solve for the missing side

$y$

$y=$ (24)

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Mute

First, seperate the two triangles.

Write out the proportions of corresponding sides to solve for $y$

$\frac{\color{#e85e00}{BC}}{\color{#e85e00}{DE}}=\frac{\color\green{y}}{\color\green{AD}}$			Definition of Similarity

$\frac{\color{#e85e00}{6}}{\color{#e85e00}{2}}=\frac{\color\green{y}}{\color\green{8}}$			Plug in the side lengths

$6 xx 8$	$=$	$2 xx y$	Cross Multiply
$48$	$=$	$2y$
$48$ $divide 2$	$=$	$2y$ $divide 2$	Divide both sides by $2$
$24$	$=$	$y$

$y=24$

Question 3 of 5

The following two triangles are similar.

Solve for the missing sides

$x$ and

$y$

$x=$ (5.88) $y=$ (5.9)

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First, seperate the two triangles.

Write out the proportions of corresponding sides to solve for $x$

$\frac{\color{#e85e00}{AB}}{\color{#e85e00}{AE}}=\frac{AC}{\color\green{AD}}$			Definition of Similarity

$\frac{\color{#e85e00}{22}}{\color{#e85e00}{5}}=\frac{x+20}{\color\green{x}}$			Plug in the side lengths

$5 xx (x+20)$	$=$	$22 xx x$	Cross Multiply
$5x +100$	$=$	$22x$	Simplify
$100$	$=$	$17x$
$100$ $divide 17$	$=$	$17x$ $divide 17$	Divide both sides by $17$
$5.88$	$=$	$x$

Do the same on corresponding sides to solve for $y$

$\frac{\color{#e85e00}{5}}{\color{#e85e00}{22}}=\frac{\color{#004ec4}{y}}{\color{#004ec4}{26}}$			Plug in the side lengths

$22 xx y$	$=$	$5 xx 26$	Cross Multiply
$22y$	$=$	$130$	Simplify
$22y$ $divide 22$	$=$	$130$ $divide 22$	Divide both sides by $22$
$y$	$=$	$5.9$

$x=5.88$ and

$y=5.9$

Question 4 of 5

The following two triangles are similar.

Solve for the missing side

$x$

$x=$ (7)

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Mute

First, seperate the two triangles.

Notice that length $AB=14$ since the sum of

$AE=6$ and

$EB=8$ is

$14$

Write out the proportions of corresponding sides to solve for $x$

$\frac{\color{#e85e00}{AE}}{\color{#e85e00}{AB}}=\frac{\color\green{ED}}{\color\green{x}}$			Definition of Similarity

$\frac{\color{#e85e00}{6}}{\color{#e85e00}{14}}=\frac{\color\green{3}}{\color\green{x}}$			Plug in the side lengths

$6 xx x$	$=$	$3 xx 14$	Cross Multiply
$6x$ $divide 6$	$=$	$42$ $divide 6$	Divide both sides by $6$
$x$	$=$	$7$

$x=7$

Question 5 of 5

The following two triangles are similar.

Solve for the missing side

$y$

$y=$ (4.8)

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First, seperate the two triangles.

Write out the proportions of corresponding sides to solve for $y$

$\frac{\color{#e85e00}{EB}}{\color{#e85e00}{BC}}=\frac{\color\green{AB}}{\color\green{y}}$			Definition of Similarity

$\frac{\color{#e85e00}{20}}{\color{#e85e00}{8}}=\frac{\color\green{12}}{\color\green{y}}$			Plug in the side lengths

$12 xx 8$	$=$	$20 xx y$	Cross Multiply
$96$	$=$	$20y$
$96$ $divide 20$	$=$	$20y$ $divide 20$	Divide both sides by $20$
$4.8$	$=$	$y$

$y=4.8$

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