13.2: tangled triangles
trace the 2 smaller triangles onto separate pieces of tracing paper.
1. turn your tracing paper and convince yourself all 3 triangles are similar.
2. write 3 similarity statements.
3. determine the scale factor for each pair of triangles.
4. determine the lengths of sides hg, gf, and hf.

Question

Accepted Answer

# Explanation:
## Step1: Identify Similar Triangles
All three triangles are right - angled (or have corresponding angles equal). Triangle $ \triangle HEG $, $ \triangle FGE $, and $ \triangle HEF $ are similar by AA (Angle - Angle) similarity criterion. In $ \triangle HEF $, $ \angle H = 53^{\circ}$, $ \angle F=37^{\circ}$, so $ \angle E = 90^{\circ}$. In $ \triangle HEG $, $ \angle H = 53^{\circ}$, $ \angle G = 90^{\circ}$, so $ \angle E=37^{\circ}$. In $ \triangle FGE $, $ \angle F = 37^{\circ}$, $ \angle G = 90^{\circ}$, so $ \angle E = 53^{\circ}$.

## Step2: Write Similarity Statements
1. $ \triangle HEG\sim\triangle HEF$ (AA similarity, $ \angle H$ is common and $ \angle HGE=\angle HEF = 90^{\circ}$)
2. $ \triangle FGE\sim\triangle HEF$ (AA similarity, $ \angle F$ is common and $ \angle FGE=\angle HEF = 90^{\circ}$)
3. $ \triangle HEG\sim\triangle FGE$ (AA similarity, $ \angle HGE=\angle FGE = 90^{\circ}$ and $ \angle HEG=\angle FGE = 37^{\circ}$ (or $ \angle H=\angle FEG = 53^{\circ}$))

## Step3: Determine Scale Factors
- For $ \triangle HEG$ and $ \triangle HEF$: The ratio of corresponding sides. In $ \triangle HEG$, side $ HE = 12$, in $ \triangle HEF$, side $ HE = 12$ and $ HF$ is the hypotenuse. Wait, actually, let's use the sides. In $ \triangle HEG$, $ HE = 12$, $ HG$ (let's find later), $ EG$ (let's find later). In $ \triangle HEF$, $ HE = 12$, $ EF = 16$, $ HF=\sqrt{12^{2}+16^{2}}=\sqrt{144 + 256}=\sqrt{400}=20$. For $ \triangle HEG$ and $ \triangle HEF$, the scale factor is $ \frac{HE}{HE}=1$? No, wait, $ \triangle HEG$ has sides $ HE = 12$, $ HG$, $ EG$. $ \triangle HEF$ has sides $ HE = 12$, $ EF = 16$, $ HF = 20$. The ratio of $ HE$ (in $ \triangle HEG$) to $ HE$ (in $ \triangle HEF$) is 1, but actually, $ \triangle HEG$ is a smaller triangle. Wait, $ \triangle HEG$ and $ \triangle FGE$: $ HE = 12$, $ EF = 16$, so the ratio of $ HE$ to $ EF$ is $ \frac{12}{16}=\frac{3}{4}$? Wait, no. Let's use the similar triangles. Since $ \triangle HEG\sim\triangle FGE$, the ratio of $ HE$ to $ FG$ (but maybe better to use the sides we know. $ \triangle HEG$ has $ HE = 12$, $ \triangle FGE$ has $ EF = 16$. Wait, actually, the scale factor between $ \triangle HEG$ and $ \triangle HEF$: $ \triangle HEG$ is similar to $ \triangle HEF$, and the ratio of $ HE$ (in $ \triangle HEG$) to $ HF$ (in $ \triangle HEF$)? No, let's take corresponding sides. In $ \triangle HEG$ and $ \triangle HEF$, $ HE$ corresponds to $ HE$, $ HG$ corresponds to $ HE$? No, I think I made a mistake. Let's use the sides:
   - $ \triangle HEG$ and $ \triangle FGE$: $ HE = 12$, $ EF = 16$, so the scale factor is $ \frac{HE}{EF}=\frac{12}{16}=\frac{3}{4}$ (or $ \frac{EF}{HE}=\frac{16}{12}=\frac{4}{3}$ depending on the order).
   - $ \triangle HEG$ and $ \triangle HEF$: The scale factor is $ \frac{HE}{HF}=\frac{12}{20}=\frac{3}{5}$ (since $ HF = 20$)? Wait, no, $ HF = 20$, $ HE = 12$, $ EF = 16$. So $ \triangle HEG$ has sides $ HE = 12$, $ HG$, $ EG$. $ \triangle HEF$ has sides $ HE = 12$, $ EF = 16$, $ HF = 20$. So the ratio of $ HE$ (in $ \triangle HEG$) to $ HE$ (in $ \triangle HEF$) is 1, but the ratio of $ HG$ (in $ \triangle HEG$) to $ HE$ (in $ \triangle HEF$)? Wait, maybe a better approach. Since $ \triangle HEG\sim\triangle FGE$, the ratio of $ HE$ to $ FG$ (but $ FG$ is equal to $ HG$? No, let's find the sides $ HG$, $ GF$, $ HF$ first.

## Step4: Determine Lengths of $ HG$, $ GF$, and $ HF$
We know that $ \triangle HEF$ is a right - triangle with $ HE = 12$, $ EF = 16$. By Pythagoras theorem, $ HF=\sqrt{HE^{2}+EF^{2}}=\sqrt{12^{2}+16^{2}}=\sqrt{144 + 256}=\sqrt{400}=20$.

Since $ \triangle HEG\sim\triangle HEF$, the ratio of corresponding sides is equal. Let $ HG=x$, $ EG = y$. Then $ \frac{HG}{HE}=\frac{HE}{HF}$ (corresponding sides of similar triangles). So $ \frac{x}{12}=\frac{12}{20}$, so $ x=\frac{12\times12}{20}=\frac{144}{20}=7.2$.

Since $ \triangle FGE\sim\triangle HEF$, let $ GF = z$, $ EG = y$. Then $ \frac{GF}{EF}=\frac{EF}{HF}$, so $ \frac{z}{16}=\frac{16}{20}$, so $ z=\frac{16\times16}{20}=\frac{256}{20}=12.8$.

And $ HF = 20$ (from Pythagoras). Also, $ HG+GF=7.2 + 12.8 = 20$, which matches $ HF = 20$.

For scale factors:
- Between $ \triangle HEG$ and $ \triangle HEF$: The ratio of sides $ \frac{HG}{HE}=\frac{7.2}{12}=\frac{3}{5}$, $ \frac{EG}{EF}=\frac{EG}{16}$ (we can find $ EG$ from $ \triangle HEG$: $ EG=\sqrt{HE^{2}-HG^{2}}=\sqrt{12^{2}-7.2^{2}}=\sqrt{144 - 51.84}=\sqrt{92.16}=9.6$, and $ \frac{9.6}{16}=\frac{3}{5}$, so scale factor is $ \frac{3}{5}$ (or $ \frac{5}{3}$ depending on the order).
- Between $ \triangle FGE$ and $ \triangle HEF$: $ \frac{GF}{EF}=\frac{12.8}{16}=\frac{4}{5}$, $ \frac{EG}{HE}=\frac{9.6}{12}=\frac{4}{5}$, so scale factor is $ \frac{4}{5}$ (or $ \frac{5}{4}$).
- Between $ \triangle HEG$ and $ \triangle FGE$: $ \frac{HG}{GF}=\frac{7.2}{12.8}=\frac{9}{16}$? Wait, no, $ \frac{HE}{EF}=\frac{12}{16}=\frac{3}{4}$, and $ \frac{HG}{EG}=\frac{7.2}{9.6}=\frac{3}{4}$, so scale factor is $ \frac{3}{4}$ (or $ \frac{4}{3}$).

# Answer:
1. All three triangles are similar by AA similarity (corresponding angles equal).
2. Similarity statements: $ \triangle HEG\sim\triangle HEF$, $ \triangle FGE\sim\triangle HEF$, $ \triangle HEG\sim\triangle FGE$
3. Scale factors:
   - Between $ \triangle HEG$ and $ \triangle HEF$: $ \frac{3}{5}$ (or $ \frac{5}{3}$)
   - Between $ \triangle FGE$ and $ \triangle HEF$: $ \frac{4}{5}$ (or $ \frac{5}{4}$)
   - Between $ \triangle HEG$ and $ \triangle FGE$: $ \frac{3}{4}$ (or $ \frac{4}{3}$)
4. Lengths: $ HG = 7.2$, $ GF = 12.8$, $ HF = 20$

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QUESTION IMAGE

Question

Explanation:

Step1: Identify Similar Triangles

Step2: Write Similarity Statements

Step3: Determine Scale Factors

Answer: