Question

finding side lengths in a right triangle what is the value of s? units

Explanation:

Step1: Use the Pythagorean theorem for \(\triangle DBC\)

In right - triangle \(DBC\), by the Pythagorean theorem \(DB^{2}+BC^{2}=DC^{2}\). Given \(DB = 15\), \(BC = 5\), then \(DC=\sqrt{15^{2}+5^{2}}=\sqrt{225 + 25}=\sqrt{250}=5\sqrt{10}\).

Step2: Prove \(\triangle ABD\cong\triangle CBD\)

Since \(\angle ABD=\angle CBD = 90^{\circ}\), \(AB = BC = 5\), and \(BD\) is common. By the Hypotenuse - Leg (HL) congruence criterion for right - triangles (\(AD = DC\) as \(AD = s\) and \(DC\) is calculated from the previous step in the right - triangle \(DBC\) and \(\triangle ABD\) and \(\triangle CBD\) are right - triangles with \(BD\) as a common leg and \(AB = BC\)). So \(s=DC\).

Answer:

\(5\sqrt{10}\)