## OGQR 2020: Question No. 87

In the figure above, if triangles ABC, ACD, and ADE are isosceles right triangles and the area of ΔABC is 6, then the area of ΔADE is:

Source | OGQR 2020 |

Type | Problem Solving |

Topic | Geometry |

Sub-Topic | Triangle |

Difficulty | Medium |

### Solution

__Given__

__Given__

In this question, we are given

- A diagram consists of three triangles ABC, ACD, and ADE – all of them are individually isosceles right triangles.
- The area of ΔABC is 6.

__To Find__

__To Find__

We need to determine

- The area of ΔADE.

__Approach & Working__

__Approach & Working__

Let us assume the length of AB = n

- As ΔABC is an isosceles right angle triangle, AB = BC = n
- Therefore, AC = √(n
^{2}+ n^{2}) = √2n

We also know that ΔACD is an isosceles right angle triangle.

- Hence, AC = CD = √2n
- Therefore, AD = √(2n
^{2}+ 2n^{2}) = 2n

Finally, ΔADE is an isosceles right-angle triangle.

- Hence, AD = DE = 2n
- Thus, the area of ΔADE = ½ * 2n * 2n = 2n
^{2}

We are also given that area of ΔABC is 6.

- Hence, we can write = ½ * n * n = ½ n
^{2}= 6- Or, n
^{2}= 12

- Or, n
- Therefore, area of ΔADE = 2n
^{2}= 2 * 12 = 24

Hence, the correct answer is option B.

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