SOLUTION: An isosceles triangle has a perimeter of 8 cm. Find a function that models its area A in terms of the length of its base b.

Question 30440: An isosceles triangle has a perimeter of 8 cm. Find a function that models its area A in terms of the length of its base b.
Answer by longjonsilver(2297) (Show Source): You can put this solution on YOUR website!
Draw an isosceles triangle and drop a perpendicular from the apex to the mid-point of the base, b. You now have 2 right-angled triangles.

Let the height = h
Let both the angled sides of the isosceles triangle be x, so that the perimeter is x+x+b --> 2x+b=8 which means that x=(8-b)/2.

What we need to do is rewrite the x in terms of b, to satisfy the question.

We have 3 things we can use:
1. Pythagoras' Theorem for one of the right-angled triangles
2. Area of a triangle is (1/2)*base*height
3. the relationship

1. In either of the 2 rightangled triangles, we have .

This means that
-- eqn1

Now, from , we have . So . Therefore

Right, we now have an equation relating x to b. If we substitute this into eqn1, we get:

so that

Right, we now have height in terms of just b.

So, from Area of a triangle is (1/2)*base*height, we get

--> this is the useful line of maths to understand. From here, we get:

I think this is a correct function for the area of the isosceles triangle when the perimeter is 8cm.

Jon.

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