Question 896764: The area of an isosceles triangle is 600 square meter. Each equal side is 13 m. Find the length of the base .
Found 2 solutions by Theo, richwmiller:
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! 600 square doesn't sound right.
there's no way you can have that.
let me explain.
a right triangle with a hypotenuse of 13 units will have legs of 5 and 12.
this is because 5^2 + 12^2 = 13^2
your isosceles triangle is composed of two right triangle back to back.
the area of each of the right triangles will be 1/2 * 5 * 12 = 1/2 * 60 = 30
multiply that by 2 and you get an area of 60 square units.
if the area of the isosceles triangle is 60 square units, then the problem can be solved.
if it's not, then the problem can't be solved.
assuming the area is 60 square units rather than 600, I would solve as follows:
area of the isosceles triangle is equal to 2 times the area of the right triangle that is formed when you drop an altitude from the vertex to the base.
each of these right triangles has 13 as a hypotgenuse. these are the equal legs of the isosceles triangle.
the area of each of these right triangles is equal to 1/2 * b * h which is equal to 30 square units.
so for each of the right triangles, the formula for the area becomes:
A = 1/2 * b * h which becomes:
30 = 1/2 * b * h
we don't know b or h, but we do know that:
b^2 + h^2 = 13^2
we solve for h in terms of b as follows:
b^2 + h^2 = 13^2
subtract b^2 from both sides of the equation to get:
h^2 = (13^2 - b^2)
take the square root of both sides of this equation to get:
h = square root of (13^2 - b^2)
formula for the area of a right triangle now becomes:
30 = 1/2 * b * h which becomes:
30 = 1/2 * b * sqrt(13^2 - b^2)
we can now solve for b as follows:
multiply both sides of this equation by 2 to get:
60 = b * sqrt(13^2 - b^2)
square both sides of this equation to get:
3600 = b^2 * (13^2 - b^2)
simplify to get:
3600 = b^2 * 169 - b^2 * b^2 which becomes:
3600 = 169*b^2 - b^4
add b^4 to both sides of this equation and subtract 169b^2 from both sides of this equation to get:
b^4 - 169b^2 + 3600 = 0
let x = b^2 and the equation becomes:
x^2 - 169x + 3600 = 0
factor this equation to get:
(x-25) * (x-144) = 0
solve for x to get;
x = 25 or x = 144
now replace x with b^2 and you get:
b^2 = 25 or b^2 = 144
solve for b to get:
b = 5 or b = 12
now that you know b, you can solve for h.
when b = 5, h = 12
when b = 12, h = 5
in both cases, the area of the right triangle is 1/2 * b * h = 1/2 * 5 * 12 = 1/2 * 60 = 30
the 2 right triangles make up the isosceles triangle and the area of the isosceles triangle is 60.
the length of the base of the isosceles triangle is 2 times the length of the base of each right triangle.
the base will therefore be either 24 or 10, depending on whether the height of the isosceles triangle is 5 or 12.
the following picture should show you what i mean.
Answer by richwmiller(17219)  (Show Source):
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