Triangle with sides 1,1,5x has same area as triangle with sides 1,1,12x. What is x

Find the exact numeric value of x.

Both triangles are isosceles (2 sides the same), so use the unique side as the base. Then you can find the height of the triangle by looking at the right triangle formed by one of the sides, the line from the top of the triangle to the midpoint of the base, and half of the base itself.

if h is the height of the first triangle and g is the height of the second, then

1^2 = h^2 + (5x/2)^2
h = sqrt(1-25x^2/4)

1^2 = g^2 + (6x)^2
g = sqrt(1-36x^2)

Area is 1/2 base times height, so,
1/2 5x h = 1/2 12x g
5h = 12g
5 sqrt(1-25x^2/4) = 12 sqrt(1-36x^2)
25(1-25x^2/4) = 144(1-36x^2)
25 - 625 x^2 / 4 = 144 - 5184 x^2
20,111 x^2/4 = 119
x^2 = 4/169
x = 2/13

Heron's formula can calculate such areas easily.

s1 = (1+1+5x)/2 = 1 + 5x/2,
s2 = (1+1+12x)/2 = 1 + 6x.

A1 = Sqrt(s1(s1-a1)(s1-b1)(s1-c1)) = Sqrt((1+5x/2)(5x/2)(5x/2)(1-5x/2)) = 5x/2 * Sqrt(1-25x^2/4)
A2 = Sqrt(s2(s2-a2)(s2-b2)(s2-c2)) = Sqrt((1 + 6x)(6x)(6x)(1-6x)) = 6x * Sqrt(1 - 36x^2).

Simplifying a little, A1 = A2 means 25/4 (1 - 25x^2/4) = 36 * (1 - 36x^2), so (5/2)^2 - (5/2)^4 x^2 = 6^2 - 6^4 x^2. This is just a quadratic equation and can easily be solved for x=2/13.

Triangle with sides 1,1,5x has same area as triangle with sides 1,1,12x.
whence 6x sqrt[1^2 -- 36x^2] = (5/2)xsqrt[1 -- 25x^2/4]
OR 36x^2 [1 -- 36x62] = (25x^2 / 4) [1 -- 25x^2/4]
OR 576 [1 -- 36x^2] = 25 [4 -- 25x^2]
Giving 20111 x^2 = 476
OR x = sqrt [ 476 / 20111] = 2 / 13 ANSWER