Question

AD is an altitude of an equilateral triangle ABC. On AD as base another equilateral triangle ADE is constructed. Prove that ar(∆MDE): ar(∆ABC)

Answer 1

Answer:

We have an equilateral △ABC in which AD is altitude. An equilateral △ADE is drawn using AD as base.

Since, the two triangle are equilateral, the two triangles will be similar also.

△ADE∼△ABC

We know that according to the theorem, the ratio of areas of two similar triangles is equal to the ratio of the squares of any two corresponding sides.

⇒ ar(△ABC)ar(△ADE)=(ABAD)2 ----- ( 1 )

Now, △ABC is an equilateral triangle.

∴ ∠B=60o

⇒ sinB=ABAD

⇒ sin60o=ABAD

⇒ 23=ABAD

⇒ (ABAD)2=43

Substituting above value in equation ( 1 ) we get,

⇒ ar(△ABC)ar(△ADE)=43 ------ Hence proved.