A geometry problem by Ajay Sambhriya

Geometry Level 3

The figure above shows an equilateral triangle ABC is inscribed inside a circle of radius 5.

MN is a chord of length 6 and is parallel to AB.

MN cuts the triangle at points D and E.

Find the area of the triangle DEC to 2 decimal places.

1 solution

Marta Reece
May 17, 2017

$ON=5, NP=3$

$OP=\sqrt{5^2-3^2}=4, PC=1$

The triangle $ABC$ is equilateral, therefore $\triangle DCE$ is also.

If the height of an equilateral triangle is $1$ , that the base of it is

$b=2\times1\times \tan30^\circ=\dfrac{2}{\sqrt{3}}$

Area of $\triangle DCE=\dfrac12\times \dfrac{2}{\sqrt{3}}\times 1=\dfrac{1}{\sqrt{3}}\approx \boxed{0.577}$