Question

unit 1 similarity
assignment 1d – similar triangle applications
name: ______
per. ____

1. △klm ~ △rst find st.

diagrams of triangles klm and rst with side lengths 72, 88 (klm) and 45, 50 (rst), and work shown: ( \frac{st}{lk} = \frac{sr}{lm} ), ( 88 cdot x = 45 cdot 72 ), ( 88x = 3240 )
st=______

1. △efg ~ △tuv find the value of x.

diagrams of triangles efg (sides 56, 84, angle 86°) and tuv (side 12, ( 2x - 14 ), angle 86°), and work shown: ( \frac{ut}{ef} = \frac{tv}{eg} ), ( 84(2x - 14) = 12 cdot 56 ), ( 168x - 1176 = 672 ), etc.
x=______

1. △pqr ~ △jkl if ( mangle j = 56^circ ) and ( mangle k = 48^circ ) what is ( mangle r )?

diagrams of triangles jkl and pqr
( mangle r )=______

1. △gfe ~ △nml. find the value of x.

diagrams of triangles gfe (sides 44, ( 9x + 4 )) and nml (sides 11, 10)
x=______

Explanation:

Problem 1: $\triangle KLM \sim \triangle RST$, Find $ST$

Step 1: Identify Corresponding Sides

Since $\triangle KLM \sim \triangle RST$, the corresponding sides are proportional. So, $\frac{ST}{LK} = \frac{SR}{LM}$. From the diagram, $LK = 72$, $SR = 45$, $LM = 88$, and $ST = x$ (let's denote $ST$ as $x$).

Step 2: Set Up Proportion

Substitute the values into the proportion: $\frac{x}{72} = \frac{45}{88}$.

Step 3: Solve for $x$

Cross - multiply: $88x = 45\times72$. Calculate $45\times72 = 3240$. Then, $x=\frac{3240}{88}=\frac{810}{22}=\frac{405}{11}\approx36.82$? Wait, no, wait, maybe I mixed up the corresponding sides. Wait, let's re - check. In similar triangles, the order of the letters matters. $\triangle KLM \sim \triangle RST$, so $KL$ corresponds to $RS$, $LM$ corresponds to $ST$, and $KM$ corresponds to $RT$? Wait, maybe the correct proportion is $\frac{ST}{LM}=\frac{SR}{LK}$. Wait, the original student's work had $\frac{ST}{LK}=\frac{SR}{LM}$, let's follow that. So $ST\times LM=SR\times LK$. So $x\times88 = 45\times72$. Then $x=\frac{45\times72}{88}=\frac{45\times9}{11}=\frac{405}{11}\approx36.82$? But maybe the correct correspondence is $KL$ (length 72) corresponds to $RS$ (length 45), $LM$ (length 88) corresponds to $ST$ (length $x$)? Wait, no, similarity ratio: if $\triangle KLM \sim \triangle RST$, then $\frac{KL}{RS}=\frac{LM}{ST}=\frac{KM}{RT}$. So $KL = 72$, $RS = 45$, $LM = 88$, $ST = x$. So $\frac{72}{45}=\frac{88}{x}$. Cross - multiply: $72x=45\times88$. $45\times88 = 3960$. Then $x=\frac{3960}{72}=55$. Ah, I see, I had the proportion reversed. The correct proportion is based on the order of the similar triangles. So $\triangle KLM \sim \triangle RST$ means $K$ corresponds to $R$, $L$ corresponds to $S$, $M$ corresponds to $T$. So $KL$ (side between $K$ and $L$) corresponds to $RS$ (side between $R$ and $S$), $LM$ (side between $L$ and $M$) corresponds to $ST$ (side between $S$ and $T$), $KM$ (side between $K$ and $M$) corresponds to $RT$ (side between $R$ and $T$). So $\frac{KL}{RS}=\frac{LM}{ST}$. So $KL = 72$, $RS = 45$, $LM = 88$, $ST = x$. So $\frac{72}{45}=\frac{88}{x}$. Cross - multiply: $72x = 45\times88$. $45\times88=45\times(80 + 8)=45\times80+45\times8 = 3600+360 = 3960$. Then $x=\frac{3960}{72}=55$.

Step 1: Identify Corresponding Sides

Since $\triangle EFG \sim \triangle TUV$, the corresponding sides are proportional. From the diagram, $EF = 56$, $EG = 84$, $TU=2x - 14$, $TV = 12$. The proportion is $\frac{TU}{EF}=\frac{TV}{EG}$.

Step 2: Set Up Proportion

Substitute the values: $\frac{2x - 14}{56}=\frac{12}{84}$.

Step 3: Simplify the Proportion

Simplify $\frac{12}{84}=\frac{1}{7}$. So the equation becomes $\frac{2x - 14}{56}=\frac{1}{7}$. Cross - multiply: $7(2x - 14)=56\times1$.

Step 4: Solve for $x$

Expand the left - hand side: $14x-98 = 56$. Add 98 to both sides: $14x=56 + 98=154$. Divide both sides by 14: $x=\frac{154}{14}=11$.

Step 1: Use Similar Triangles Angle Correspondence

In similar triangles, corresponding angles are equal. So $\angle P\cong\angle J$, $\angle Q\cong\angle K$, $\angle R\cong\angle L$.

Step 2: Use Triangle Angle Sum Theorem

The sum of the interior angles of a triangle is $180^{\circ}$. In $\triangle JKL$, $m\angle J + m\angle K+m\angle L=180^{\circ}$. We know $m\angle J = 56^{\circ}$, $m\angle K = 48^{\circ}$, so $m\angle L=180-(56 + 48)=180 - 104 = 76^{\circ}$.

Step 3: Find $m\angle R$

Since $\angle R\cong\angle L$, $m\angle R=m\angle L = 76^{\circ}$.

Answer:

$ST = 55$

Problem 2: $\triangle EFG \sim \triangle TUV$, Find $x$