Question

if $qr = 13$, $rs = 10$, $qs = 16$, $uv = 11$, and $tv = 17.6$, find the perimeter of $\triangle tuv$. round your answer to the nearest tenth if necessary. figures are not necessarily drawn to scale.

Explanation:

Step1: Identify Similar Triangles

Triangles \( \triangle QRS \) and \( \triangle TUV \) have the same angle measures (\( 58^\circ \), \( 83^\circ \), \( 39^\circ \)), so they are similar by AA (Angle - Angle) similarity criterion.

Step2: Find the Scale Factor

First, find the perimeter of \( \triangle QRS \). The sides of \( \triangle QRS \) are \( QR = 13 \), \( RS = 10 \), \( QS = 16 \). So the perimeter of \( \triangle QRS \), \( P_{QRS}=13 + 10+16=39 \).
Let the scale factor be \( k \). We can find the scale factor by comparing the corresponding sides. Let's assume \( TV \) corresponds to \( QR \) (since the angles are equal, we can match the sides). So \( k=\frac{TV}{QR}=\frac{17.6}{13}\approx1.3538 \).

Step3: Find the Perimeter of \( \triangle TUV \)

For similar triangles, the ratio of perimeters is equal to the scale factor. Let \( P_{TUV} \) be the perimeter of \( \triangle TUV \). We know that \( \frac{P_{TUV}}{P_{QRS}}=k \). Also, we can check another pair of sides. Let's take \( UV = 11 \) and assume it corresponds to \( RS = 10 \). Then \( k=\frac{UV}{RS}=\frac{11}{10} = 1.1 \). Wait, there is a mistake here. Wait, let's re - check the angle correspondence.
In \( \triangle QRS \), angles are \( \angle S = 58^\circ \), \( \angle R=83^\circ \), \( \angle Q = 39^\circ \). In \( \triangle TUV \), angles are \( \angle V = 58^\circ \), \( \angle U = 83^\circ \), \( \angle T=39^\circ \). So the correspondence is \( \triangle QRS\sim\triangle VUT \) (angle - angle - angle). So \( QR \) corresponds to \( VT \), \( RS \) corresponds to \( UT \), and \( QS \) corresponds to \( UV \).
So \( \frac{QS}{UV}=\frac{QR}{VT}=\frac{RS}{UT} \). Given \( QS = 16 \), \( UV = 11 \), so the scale factor \( k=\frac{QS}{UV}=\frac{16}{11}\approx1.4545 \). Wait, no, \( UV = 11 \), \( QS = 16 \), so if \( UV \) corresponds to \( QS \), then \( k=\frac{UV}{QS}=\frac{11}{16}=0.6875 \)? No, this is confusing. Wait, let's use the correct angle - side correspondence.
Since \( \angle Q = 39^\circ \), \( \angle T = 39^\circ \); \( \angle R = 83^\circ \), \( \angle U = 83^\circ \); \( \angle S = 58^\circ \), \( \angle V = 58^\circ \). So the correspondence is \( Q
ightarrow T \), \( R
ightarrow U \), \( S
ightarrow V \). So \( QR
ightarrow TU \), \( RS
ightarrow UV \), \( SQ
ightarrow VT \).
So \( RS = 10 \), \( UV = 11 \), so the scale factor \( k=\frac{UV}{RS}=\frac{11}{10}=1.1 \).
\( QR = 13 \), so \( TU=k\times QR=1.1\times13 = 14.3 \)
\( QS = 16 \), so \( VT=k\times QS=1.1\times16 = 17.6 \) (which matches the given \( TV = 17.6 \), wait \( VT \) is \( TV \)? Maybe a typo, \( VT = TV \)).
Now, the perimeter of \( \triangle TUV \) is \( TU+UV + TV=14.3+11 + 17.6=42.9 \). Wait, let's recalculate the perimeter of \( \triangle QRS \): \( 13 + 10+16 = 39 \). The scale factor \( k=\frac{11}{10}=1.1 \), so the perimeter of \( \triangle TUV=39\times1.1 = 42.9 \). Let's verify with the sides:
If \( RS = 10 \) (corresponds to \( UV = 11 \), scale factor 1.1), \( QR = 13 \) (corresponds to \( TU=13\times1.1 = 14.3 \)), \( QS = 16 \) (corresponds to \( TV = 16\times1.1=17.6 \), which matches the given \( TV = 17.6 \)). So the perimeter of \( \triangle TUV \) is \( 14.3+11 + 17.6 = 42.9 \).

Answer:

\( 42.9 \)