The perimeter of an isosceles tri angle is 64 cm and each of the equal sides is `5/6` times the base. What is the area (in `cm^2`) of the triangle?

Options

A. 169
B. 192
C. 196
D. 184

Answer

To solve the problem step by step, we will follow the logic outlined in the video transcript.

### Step 1: Understand the Given Information
We know that the perimeter of the isosceles triangle is 64 cm, and each of the equal sides is \( \frac{5}{6} \) times the base.

### Step 2: Define Variables
Let:
- \( x \) = base of the triangle
- Each equal side = \( \frac{5}{6}x \)

### Step 3: Set Up the Perimeter Equation
The perimeter \( P \) of the triangle can be expressed as:
\[ P = \text{base} + \text{equal side} + \text{equal side} = x + \frac{5}{6}x + \frac{5}{6}x \]
This simplifies to:
\[ P = x + \frac{10}{6}x = x + \frac{5}{3}x = \frac{8}{3}x \]

### Step 4: Solve for the Base
Since the perimeter is given as 64 cm, we can set up the equation:
\[ \frac{8}{3}x = 64 \]
To solve for \( x \), multiply both sides by \( \frac{3}{8} \):
\[ x = 64 \times \frac{3}{8} \]
Calculating this gives:
\[ x = 64 \times 0.375 = 24 \text{ cm} \]

### Step 5: Calculate the Length of Equal Sides
Now, we can find the length of each equal side:
\[ \text{Equal side} = \frac{5}{6}x = \frac{5}{6} \times 24 = 20 \text{ cm} \]

### Step 6: Identify the Triangle Dimensions
Now we have:
- Base \( x = 24 \) cm
- Each equal side = 20 cm

### Step 7: Find the Height of the Triangle
To find the area, we need the height. We can drop a perpendicular from the apex of the triangle to the base, which will bisect the base into two equal segments of \( \frac{x}{2} = \frac{24}{2} = 12 \) cm.

Using the Pythagorean theorem:
\[ \text{(Equal side)}^2 = \text{(Height)}^2 + \text{(Half base)}^2 \]
Substituting the values:
\[ 20^2 = h^2 + 12^2 \]
This simplifies to:
\[ 400 = h^2 + 144 \]
Subtracting 144 from both sides:
\[ h^2 = 256 \]
Taking the square root:
\[ h = 16 \text{ cm} \]

### Step 8: Calculate the Area of the Triangle
Now we can calculate the area \( A \) using the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
Substituting the values:
\[ A = \frac{1}{2} \times 24 \times 16 \]
Calculating this gives:
\[ A = 12 \times 16 = 192 \text{ cm}^2 \]

### Final Answer
The area of the triangle is \( 192 \text{ cm}^2 \).

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