Question

Given That An Isosceles Triangle Has 2 Equal Sides 5cm And An Area Of 12cm2 .Find The Base Of The Triangle.

Answer 1

Cut your isosceles triangle in two equals right-angled triangles, and consider one of them.

(A) Its area is 12/2 = 6 cm².

(B) Its hypotenuse length is 5 cm.

Let us call x and y the other sides, x being half the base of the initial triangle, and y the height from the base . We have :

(A) x × y / 2 = 6, 
So, x × y = 3 × 4
                = 12 

(B) x² + y² = 5² = 25    ...from Pythagoras Theorem 

Now, There are many ways to solve this simple quadratic system,

One of them is to recognize the “Caconical” Pythagorean example 3^2 + 4^2 = 5^2,

Leading to the two solutions (x = 3 and y = 4), or (x = 4 and y = 3).

The isosceles triangle base length is 2 × x, so it is 6 or 8.

There’s another method too:

Let the base be b cms.

Then the semi-perimeter s = (b+5+5)/2 = (b+10)/2.

By Heron's formula, area of the triangle = rt {s(s-a)(s-b)(s-c)}

i.e., 12 = rt{(b+10)/2 ((b+10)/2 -b))((b+10)/2 -5)((b+10)/2 -5))

Squaring, 144 = (b+10)/2 (10-b)/2 (b/2)(b/2)

Simplifying we get b^4 -100b^2 + 2304 = 0 which yields b = 6 or 8.

Hence the length of the base is 6 cms or 8 cms.

Hope This Helps :)