Question

A composite geometrical shape is made up of a square, equilateral and right triangles. Calculate the area of hatched triangle?

Answer 1

`156.25`

Explanation:

As `triangleABD` is a right triangle, we have, using the Pythagorean theorem, `bar(BD)=sqrt(25^2+25^2)=25sqrt(2)`. Because `triangleBDE` is equilateral, this implies that `bar(ED)=25sqrt(2)`.

Next, we can note that `triangleABD` is a `45-45-90` right triangle, meaning the angle `angleADB = 45^@`. Additionally, as `triangleBDE` is equilateral, we have `angleBDE=60^@`. Using those facts, along with `angleADF=180^@`, we have:

`angleEDF = angleADF - angleBDE-angleABD`

`=180^@-60^@-45^@`

`=75^@`

As `triangleEDF` is a right triangle, we can use the sine and cosine functions to say

`bar(EF)/bar(ED)=sin(75^@)`

`=>bar(EF)=bar(ED)sin(75^@)=25sqrt(2)sin(75^@)`

and

`bar(DF)/bar(ED)=cos(75^@)`

`=>bar(DF)=bar(ED)cos(75^@)=25sqrt(2)cos(75^@)`

As `bar(DF)` and `bar(EF)` are the base and height, respectively, of the hatched triangle, we can use the formula `A=1/2bh` to obtain the area of the hatched triangle as

`A =1/2bar(DF)*bar(EF)`

`=1/2(25sqrt(2)cos(75^@))(25sqrt(2)sin(75^@))`

`=625cos(75^@)sin(75^@)`

`=156.25`