Question

what is the length of side x? triangle with hypotenuse 13, base 12, and height x

Explanation:

Step1: Identify the triangle type

The triangle is a right - triangle (one angle is a right angle as it has two perpendicular sides). We can use the Pythagorean theorem, which states that for a right - triangle with hypotenuse \(c\) and the other two sides \(a\) and \(b\), \(a^{2}+b^{2}=c^{2}\). Here, the hypotenuse \(c = 13\) and one of the legs \(b = 12\), and the other leg is \(x\).

Step2: Apply the Pythagorean theorem

We know that \(x^{2}+12^{2}=13^{2}\). First, calculate \(12^{2}=144\) and \(13^{2}=169\). Then, we can rewrite the equation as \(x^{2}=13^{2}-12^{2}\). Substituting the values, we get \(x^{2}=169 - 144\).

Step3: Solve for \(x\)

Calculate \(169-144 = 25\). So, \(x^{2}=25\). Taking the square root of both sides, we have \(x=\sqrt{25}=5\) (we take the positive square root since length cannot be negative).

Answer:

\(5\)