In the following figure, A D = A B. Also ∠ D A B =...

Baladdaa9

Baladdaa9

Answered

2022-07-22

In the following figure, A D = A B. Also D A B = D C B = A E C = 90 and A E = 5. Find the area of quadrilateral ABCD.

( x 2 25 )

Answer & Explanation

Kali Galloway

Kali Galloway

Expert

2022-07-23Added 16 answers

Step 1
Join AC and hence see by cyclic quad.
A C E = A C B = 45 o
Hence E C = 5 , A C = 5 2
Step 2
Thus, 2 x 2 = ( x 2 25 + 5 ) 2 + B C 2
Now apply cosine rule in Δ A B C to get
cos 45 = x 2 50 B C 2 10 2 B C
Francisco Proctor

Francisco Proctor

Expert

2022-07-24Added 6 answers

Step 1
Take B D C = θ, so D E = x c o s ( θ + 45 ) , x s i n ( θ + 45 ) = 5 , E C = x 2 c o s θ x c o s ( θ + 45 ) , D C = x 2 c o s θ , B C = x 2 s i n θ
Δ A D B + Δ B D C = Δ A D E + A E C B
Δ A D B + Δ B D C = x 2 2 + x 2 s i n θ c o s θ = x 2 2 ( s i n θ + c o s θ ) 2
Step 2
Δ A D E + A E C B = x 2 s i n θ c o s θ x 2 2 s i n θ c o s ( θ + 45 ) + 5 2 x c o s θ
Equating both we get x = 5 2 ( s i n θ + c o s θ ) .
Using this in x 2 2 ( s i n θ + c o s θ ) 2 we get area = 25.

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