The perimeter of an isosceles triangle is 270 and the side is 75. Find the area of the triangle?

1. Vertices of the triangle A, B, C. AB = BC = 75 units.

2. We calculate the length of the base of the speaker:

AC = 270 – 75 – 75 = 120 units.

3. From the top B we draw the height of the HB to the base of the AC.

4. According to the properties of an isosceles triangle, the BH height also serves as a median, that is, it divides the side to which it is drawn into two identical segments.

Therefore AH = CH = AC: 2 = 120: 2 = 60 units.

5. We calculate the length of the height BH, which is the leg of the right-angled triangle ABH.

For the calculation, we use the Pythagorean theorem:

BH = √AB² – AH² = √75² – 60² = √5625 – 3600 = √2025 = 45 units.

6. Calculate the area (S) of a given triangle:

S = АС х ВН / 2 = 120 х 45/2 = 2700 units of measurement².