Answer:
The length of a leg is 12.5 and its base is 15
Explanation:
Using Heron's formula, the area of the triangle A = √[s(s - a)(s - b)(s - c)] where s = (a + b + c)/2. Now, for an isosceles, a = b and c = its base.
So, A = √[s(s - a)(s - a)(s - c)] = √[s(s - a)²(s - c)] = (s - a)√[s(s - c)] and s = (a + a + c)/2 = a + c/2 ⇒ s = a + c/2 (1)
Given that the perimeter a + b + c = 2a + c = 40 = 2s ⇒ s = 40/2 = 20 and A = hc/2 where h = length of altitude to base = 10.
So, A = 10c/2 = 5c
So, 5c = (s - a)√[s(s - c)]
From (1) s - a = c/2.
So, 5c = (s - a)√[s(s - c)]
5c = (c/2)√[s(s - c)]
10 = √[s(s - c)]
squaring by sides, we have
100 = s(s - c) since s = 20,
s(s - c) = 100
20(20 - c) = 100
20 - c = 100/20
20 - c = 5
c = 20 - 5
c = 15
From (1),
s = a + c/2
a = s - c/2
= 20 - 15/2
= 20 - 7.5
= 12.5
Since a = b = 12.5
So, the length of a leg is 12.5 and its base is 15