Answer:
a= 22.5
b= 37.5
Explanation:
In △BCD:
Applying Pythagoras' Theorem,
a² +30²= b²
a² +900= b² -----(1)
In △ABC:
Applying Pythagoras' Theorem,
b² +50²= (40 +a)² -----(2)
Substitute (1) into (2):
a² +900 +50²= 40² +2(40)(a) +a² (expand bracket)
a² +900 +2500= 1600 +80a +a²
a² +3400= a² +80a +1600 (simplify)
a² +3400 -a² -80a -1600= 0 (bring everything to 1 side)
-80a +1800= 0
80a= 1800 (+80 on both sides)
a= 1800 ÷80
a= 22.5
Subst. a= 22.5 into (1):
22.5² +900= b²
b²= 506.25 +900
b²= 1406.25
b= √1406.25 (square root both sides)
b= 37.5 (reject negative value since b is a length)
☆(a +b)²= a² +2ab +b²