ABC and XYZ are two similar triangles with Angle Z = Angle C whose areas are respectively `32 and 60.5`. If XY = 7.7 cm,then what is AB equal to?

Options

A. 5.6 cm
B. 5.8 cm
C. 6.0 cm
D. 6.2 cm

Answer

To find the length of side AB in triangle ABC, given that triangles ABC and XYZ are similar, we can follow these steps:

### Step 1: Understand the relationship between the areas of similar triangles
For two similar triangles, the ratio of their areas is equal to the square of the ratio of their corresponding sides. This can be expressed as:
\[ \frac{\text{Area of } ABC}{\text{Area of } XYZ} = \left(\frac{AB}{XY}\right)^2 \]

### Step 2: Substitute the given values
From the problem, we know:
- Area of triangle ABC = 32
- Area of triangle XYZ = 60.5
- Length of XY = 7.7 cm

Substituting these values into the equation gives us:
\[ \frac{32}{60.5} = \left(\frac{AB}{7.7}\right)^2 \]

### Step 3: Cross-multiply to eliminate the fraction
Cross-multiplying gives:
\[ 32 \cdot (7.7)^2 = 60.5 \cdot (AB)^2 \]

### Step 4: Calculate \( (7.7)^2 \)
Calculating \( (7.7)^2 \):
\[ (7.7)^2 = 59.29 \]

### Step 5: Substitute back into the equation
Now substituting back into the equation:
\[ 32 \cdot 59.29 = 60.5 \cdot (AB)^2 \]

### Step 6: Calculate \( 32 \cdot 59.29 \)
Calculating \( 32 \cdot 59.29 \):
\[ 32 \cdot 59.29 = 1896.88 \]

### Step 7: Set up the equation for \( AB^2 \)
Now we have:
\[ 1896.88 = 60.5 \cdot (AB)^2 \]

### Step 8: Solve for \( (AB)^2 \)
Dividing both sides by 60.5:
\[ (AB)^2 = \frac{1896.88}{60.5} \]

### Step 9: Calculate \( \frac{1896.88}{60.5} \)
Calculating this gives:
\[ (AB)^2 \approx 31.36 \]

### Step 10: Take the square root to find \( AB \)
Taking the square root of both sides:
\[ AB \approx \sqrt{31.36} \approx 5.6 \text{ cm} \]

### Final Answer
Thus, the length of side AB is approximately:
\[ \boxed{5.6 \text{ cm}} \]