The question asks:

Q5. The base of an isosceles triangle measures 24 cm and its area is 84 sq. cm. Find its perimeter.

Let me solve it step by step.

Step 1: Formula for the area of a triangle

The formula for the area of a triangle is:

$\text{Area} = \frac{1}{2} \times \text{Base} \times \text{Height}$

From the question:

• Base = 24 cm
• Area = 84 cm²

Let’s substitute the values into the formula to find the height:

$84 = \frac{1}{2} \times 24 \times \text{Height}$

Simplify:

$84 = 12 \times \text{Height}$

$\text{Height} = \frac{84}{12} = 7 \, \text{cm}$

Step 2: Determine the length of the equal sides

In an isosceles triangle, the two equal sides form the hypotenuse of two right triangles when we draw a perpendicular from the vertex opposite the base to the midpoint of the base. This height divides the base into two equal parts.

• Half of the base = $\frac{24}{2} = 12 \, \text{cm}$
• Height = $7 \, \text{cm}$

Using the Pythagorean theorem to find the length of the equal sides ($L$):

$L^2 = \left(\frac{\text{Base}}{2}\right)^2 + \text{Height}^2$

Substitute the values:

$L^2 = 12^2 + 7^2$

$L^2 = 144 + 49 = 193$

$L = \sqrt{193} \, \text{cm}$

Step 3: Calculate the perimeter

The perimeter of a triangle is the sum of all its sides:

$\text{Perimeter} = \text{Base} + 2 \times L$

Substitute the values:

$\text{Perimeter} = 24 + 2 \times \sqrt{193}$

Approximating $\sqrt{193} \approx 13.89$:

$\text{Perimeter} \approx 24 + 2 \times 13.89 = 24 + 27.78 = 51.78 \, \text{cm}$

Final Answer:

The perimeter of the isosceles triangle is approximately:

$\boxed{51.78 \, \text{cm}}$

Would you like further details or have any questions? Here are some related questions to explore:

1. What is the Pythagorean theorem, and how is it used in solving triangle problems?
2. How do you calculate the height of a triangle from its area and base?
3. Why does the perpendicular height divide the base equally in an isosceles triangle?
4. How would the perimeter change if the base were different but the area stayed the same?
5. How do you simplify square roots like $\sqrt{193}$ more accurately?

Tip: In isosceles triangles, always use symmetry to simplify calculations involving height and equal sides!