Mathematics Question and Solution
Let the two congruent angles be a.
Let the base of the ascribed triangle be b.
tana = 8/b
b = 8/(tana) ----- (1).
tan2a = 18/b
b = 18/(tan2a) ----- (2).
Equating (1) and (2).
8/(tana) = 18/(tan2a)
And tan2a = 2tana/(1-tan²a)
Therefore;
8/(tana) = 18/(2tana/(1-tan²a))
Let p be tana.
8/p = 18/(2p/(1-p²))
8/p = 18(1-p²)/2p
Cross Multiply.
16p = 18p(1-p²)
16p = 18p-18p³
16 = 18-18p²
18p² =2
p² = ⅑
p = ⅓
And p = tana
tana = ⅓
a = atan(⅓)
a = 18.43494882292°
Calculating b, base of the triangle.
tan18.43494882292 = 8/b
b = 24 units.
Therefore Shaded Area Blue is;
½*10*24
= ½(240)
= 120 square units.
