Question

from a point in the interior of an equilateral triangle, perpendiculars are drawn on the three sides. the lengths of the perpendiculars are 14 cm, 10 cm and 6 cm. find the area of the triangle.

Answer 1

The area of the equilateral triangle is 300√3 square centimeters.

Let's solve this problem step-by-step:

1. Relate the perpendiculars to the side length:

Call the point inside the triangle O and the feet of the perpendiculars P, Q, and R.

Since OP, OQ, and OR are perpendicular to the sides of the equilateral triangle, they divide the triangle into three smaller triangles OAP, OBQ, and OCR.

Each of these smaller triangles is a right triangle with one leg being half the side length of the equilateral triangle (due to the perpendicular bisector theorem) and the other leg being the corresponding perpendicular length (14 cm, 10 cm, and 6 cm).

2. Find the side length of the equilateral triangle:

Let the side length of the equilateral triangle be a.

Using the Pythagorean theorem in each of the smaller triangles:

OAP: a^2/4 + 14^2 = a^2 --> a^2 = 392 --> a = 20√3 cm

3. Calculate the area of the equilateral triangle:

The area of an equilateral triangle is given by √3/4 * side^2.

Therefore, the area of triangle ABC is √3/4 * (20√3)^2 = √3/4 * 1600 = 300√3 square centimeters.