Question

find the length of the missing side of the right triangle.

Explanation:

Step1: Apply Pythagorean theorem

For a right - triangle with legs \(a = 15\), \(c=39\) (assuming \(c\) is the hypotenuse), the Pythagorean theorem is \(a^{2}+b^{2}=c^{2}\). We need to solve for \(b\). So, \(b=\sqrt{c^{2}-a^{2}}\).

Step2: Substitute values

Substitute \(a = 15\) and \(c = 39\) into the formula: \(b=\sqrt{39^{2}-15^{2}}=\sqrt{(39 + 15)(39 - 15)}\) (using \(x^{2}-y^{2}=(x + y)(x - y)\)).
First, calculate \((39 + 15)=54\) and \((39 - 15)=24\). Then \(b=\sqrt{54\times24}\).
Factorize: \(54=2\times3^{3}\), \(24 = 2^{3}\times3\). So \(54\times24=2^{4}\times3^{4}\).
Then \(b=\sqrt{2^{4}\times3^{4}}=2^{2}\times3^{2}\).

Step3: Calculate the result

\(2^{2}\times3^{2}=4\times9 = 36\).

Answer:

\(36\)