The ratio of three sides of a triangle 5:5:8. If the area of triangle is `12cm^(2)`, then what is the length (in cm.) of the equal sides ?

Options

A. 5
B. 8
C. 6
D. `2.5`

Answer

To find the length of the equal sides of the triangle with sides in the ratio 5:5:8 and an area of 12 cm², we can follow these steps:

### Step 1: Define the sides of the triangle
Let the lengths of the sides of the triangle be:
- Side 1 = 5x
- Side 2 = 5x
- Side 3 = 8x

### Step 2: Identify the type of triangle
Since two sides are equal (5x and 5x), this is an isosceles triangle.

### Step 3: Use the area formula for the triangle
The area of a triangle can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
In this case, we can take the base as the side of length 8x.

### Step 4: Find the height using Pythagoras' theorem
To find the height, we can drop a perpendicular from the vertex opposite the base (8x) to the base. This will bisect the base into two equal parts of length \(4x\).

Using the Pythagorean theorem in one of the right triangles formed:
\[ AD^2 + BD^2 = AB^2 \]
Where:
- \(AD\) is the height (h)
- \(BD\) is half the base (4x)
- \(AB\) is the equal side (5x)

Substituting the values:
\[ h^2 + (4x)^2 = (5x)^2 \]
\[ h^2 + 16x^2 = 25x^2 \]
\[ h^2 = 25x^2 - 16x^2 \]
\[ h^2 = 9x^2 \]
\[ h = 3x \]

### Step 5: Substitute the height into the area formula
Now, substituting the height back into the area formula:
\[ 12 = \frac{1}{2} \times (8x) \times (3x) \]
\[ 12 = 12x^2 \]

### Step 6: Solve for x
To find x, we divide both sides by 12:
\[ x^2 = 1 \]
\[ x = 1 \]

### Step 7: Calculate the length of the equal sides
Now, substituting the value of x back into the lengths of the equal sides:
\[ \text{Length of equal sides} = 5x = 5 \times 1 = 5 \text{ cm} \]

### Final Answer
The length of the equal sides is **5 cm**.

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