2) $$ riangle P S E- riangle T S V \ln riangle P S E, P S=4.5 \mathrm{~cm}, S E-52 \mathrm{~cm}, P E-54 \mathrm{~cm} $$ and $$ P S=T S=5: 3 $$. Construct $$ riangle P S E $$ and $$ riangle T S V $$.

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2) $$ 	riangle P S E-	riangle T S V \ln 	riangle P S E, P S=4.5 \mathrm{~cm}, S E-52 \mathrm{~cm}, P E-54 \mathrm{~cm} $$ and $$ P S=T S=5: 3 $$. Construct $$ 	riangle P S E $$ and $$ 	riangle T S V $$.

UpStudy AI · Accepted Answer

To construct the triangles $$ \triangle PSE $$ and $$ \triangle TSV $$ given the information:

- $$ \triangle PSE \sim \triangle TSV $$ (they are similar)
- $$ PS = 4.5 $$ cm, $$ SE = 52 $$ cm, $$ PE = 54 $$ cm
- The ratio of similarity $$ \frac{PS}{TS} = \frac{5}{3} $$

Follow these step-by-step instructions:

### **1. Determine the Scale Factor**
Since $$ \frac{PS}{TS} = \frac{5}{3} $$, the scale factor $$ k $$ from $$ \triangle PSE $$ to $$ \triangle TSV $$ is $$ \frac{3}{5} $$.

### **2. Calculate the Sides of $$ \triangle TSV $$**
Apply the scale factor to each corresponding side of $$ \triangle PSE $$:

- **TS:** 
  $$
  TS = \frac{3}{5} \times PS = \frac{3}{5} \times 4.5\, \text{cm} = 2.7\, \text{cm}
  $$
  
- **SV:** 
  $$
  SV = \frac{3}{5} \times SE = \frac{3}{5} \times 52\, \text{cm} = 31.2\, \text{cm}
  $$
  
- **TV:** 
  $$
  TV = \frac{3}{5} \times PE = \frac{3}{5} \times 54\, \text{cm} = 32.4\, \text{cm}
  $$

### **3. Construct $$ \triangle PSE $$**
1. **Draw Side PS:**
   - Use a ruler to draw segment $$ PS $$ of length 4.5 cm.

2. **Construct Angle at P:**
   - At point $$ P $$, use a protractor to draw the desired angle for $$ \triangle PSE $$.

3. **Draw Side SE:**
   - Along one ray from $$ P $$, measure and mark point $$ E $$ such that $$ SE = 52 $$ cm.

4. **Complete the Triangle:**
   - Connect points $$ S $$ and $$ E $$ to form $$ \triangle PSE $$.
   - Verify that $$ PE = 54 $$ cm using a ruler.

### **4. Construct $$ \triangle TSV $$**
1. **Draw Side TS:**
   - Use a ruler to draw segment $$ TS $$ of length 2.7 cm.

2. **Recreate the Corresponding Angle:**
   - At point $$ T $$, replicate the angle used at point $$ P $$ in $$ \triangle PSE $$.

3. **Draw Side SV:**
   - Along one ray from $$ T $$, measure and mark point $$ V $$ such that $$ SV = 31.2 $$ cm.

4. **Complete the Triangle:**
   - Connect points $$ S $$ and $$ V $$ to form $$ \triangle TSV $$.
   - Verify that $$ TV = 32.4 $$ cm using a ruler.

### **5. Verify Similarity**
After constructing both triangles, you can verify their similarity by checking that all corresponding sides are in the ratio $$ \frac{5}{3} $$ and that all corresponding angles are equal.

### **Illustrative Diagram**

While I can't provide a visual diagram here, following the above steps with precise measurements and angles will yield two similar triangles $$ \triangle PSE $$ and $$ \triangle TSV $$ with the specified dimensions and proportionality.

---

**Summary of Side Lengths:**

| Side        | $$ \triangle PSE $$ | $$ \triangle TSV $$ |
|-------------|---------------------|---------------------|
| PS / TS     | 4.5 cm               | 2.7 cm               |
| SE / SV     | 52 cm                | 31.2 cm              |
| PE / TV     | 54 cm                | 32.4 cm              |

By carefully following these steps, you will successfully construct both triangles with the given specifications.
To construct $$ \triangle PSE $$ and $$ \triangle TSV $$: 1. **Determine Scale Factor:** $$ \frac{3}{5} $$ 2. **Calculate $$ \triangle TSV $$ Sides:** - $$ TS = 2.7 $$ cm - $$ SV = 31.2 $$ cm - $$ TV = 32.4 $$ cm 3. **Construct $$ \triangle PSE $$:** - Draw $$ PS = 4.5 $$ cm. - Draw $$ SE = 52 $$ cm. - Connect $$ S $$ and $$ E $$. 4. **Construct $$ \triangle TSV $$:** - Draw $$ TS = 2.7 $$ cm. - Draw $$ SV = 31.2 $$ cm. - Connect $$ S $$ and $$ V $$. 5. **Verify Similarity:** Both triangles are similar with a ratio of $$ \frac{5}{3} $$. This will result in two similar triangles with the specified dimensions.

2) \( \triangle P S E-\triangle T S V \ln \triangle P S E, P S=4.5 \mathrm{~cm}, S E-52 \mathrm{~cm}, P E-54 \mathrm{~cm} \) and \( P S=T S=5: 3 \). Construct \( \triangle P S E \) and \( \triangle T S V \).

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