GIVEN: ABC an isosceles triangle, AB=AC. So angle B = angle C. Base BC = 24cm,
TO FIND: The perimeter(AB + BC + AC) =?
CONSTRUCTION: AM perpendicular to BC.
Since triangle AMB is congruent to triangle AMC( by RHS congruence criterion)
So, M is the mid point of BC. So, BM = 12cm
Since area(triangle ABC) = 1/2 * BC * AM
=> ar(tri ABC) = 1/2 * 24 * AM = 192
=> AM = 192/ 12 = 16cm.
Now in right triangle AMB
AB² = AM² + BM²
=> AB² = 16² + 12²
=> AB² = 256 + 144 = 400
=> AB = √400 = 20cm
So, AC = 20cm
So, perimeter = 20+20+24 = 64cm ……….ANS

Problem:
The base of an isosceles triangle is 24 cm and its area is 192 cm². Find its perimeter.

Solution:
To find the perimeter of the isosceles triangle, we need to determine the lengths of its sides. We can start by using the given information about the base and area of the triangle.

Step 1: Identify the given information
- Base of the triangle = 24 cm
- Area of the triangle = 192 cm²

Step 2: Understand the properties of an isosceles triangle
An isosceles triangle has two equal sides and two equal angles opposite those sides. The base of the triangle is the side that is not equal to the other two.

Step 3: Determine the height of the triangle
Since we know the area of the triangle, we can use the formula for the area of a triangle:

Area = (base * height) / 2

Substituting the given values, we have:

192 = (24 * height) / 2

Simplifying the equation:

384 = 24 * height

Height = 384 / 24 = 16 cm

Step 4: Calculate the length of the equal sides
Now that we have the height, we can use the Pythagorean theorem to calculate the length of the equal sides.

In an isosceles triangle, the height bisects the base, creating two congruent right triangles. Let's call the length of one of the equal sides "a" and the height "h".

Using the Pythagorean theorem:

a² = h² + (base/2)²

Substituting the values:

a² = 16² + (24/2)²
a² = 256 + 12²
a² = 256 + 144
a² = 400

Taking the square root of both sides:

a = √400
a = 20 cm

So, the length of each equal side is 20 cm.

Step 5: Calculate the perimeter
The perimeter of a triangle is the sum of all its sides. Since we have the length of the base and the equal sides, we can calculate the perimeter:

Perimeter = Base + 2 * Equal sides
Perimeter = 24 + 2 * 20
Perimeter = 24 + 40
Perimeter = 64 cm

Therefore, the perimeter of the isosceles triangle is 64 cm.