Question

The diagram below shows a pennant in the shape of an isosceles triangle. The equal sides each measure 13. The altitude is x + 7 , and the base is 2x. What is the length of the base

Answer 1

Answer: The length of the base is 10.

Step-by-step explanation:

You can draw an isosceles triangle like the one shown attached.

An isosceles triangle has two equal sides. Then, you can divide it into two right triangles, where the base of each one of them will be:

$b'=\frac{2x}{2}=x$

The hypotenuse of each right triangle shown is 13 and the other legs are (x+7) and x, therefore you can apply the Pythagorean Theorem:

$13^2=x^2+(x+7)^2$

Simplify the expression:

$169=x^2+(x^{2}+2(x)(7)+7^2)\\169=x^{2}+x^{2}+14x+49\\0=2x^{2}+14x-120$

Use the quadratic formula, where:

$a=2\\b=14\\c=-120$

$x=\frac{-b\±\sqrt{b^2-4ac}}{2a}$

Then:

$x=\frac{-14\±\sqrt{14^2-4(2)(-120)}}{2+2}$

$x_1=5\\x_2=-12$

Choose the positive value, because lengths are not negative.

Substitute the value of x into the base 2x. Therefore, the answer is:

$b=2x=2(5)=10$