Question

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

Explanation:

Step1: Apply Geometric Mean Theorem (Right Triangle Altitude Theorem)

In a right triangle, the altitude to the hypotenuse relates the segments of the hypotenuse and the legs. For right triangle \( \triangle ABD \) and \( \triangle CBD \), we can also use the Pythagorean theorem in \( \triangle ABD \) and \( \triangle CBD \), but more straightforwardly, first find \( AD \) using the geometric mean or directly apply Pythagorean theorem in \( \triangle ABD \) and then in \( \triangle ACD \). Wait, actually, \( \triangle ABD \) is right-angled at \( B \), so \( AD^2 = AB^2 + BD^2 \). Wait, no, wait: \( AB = 8 \), \( BD = 15 \), so \( AD^2 = 8^2 + 15^2 \). Wait, but also, \( BC = 5 \), \( BD = 15 \), \( AB = 8 \), and \( AC = AB + BC = 8 + 5 = 13 \)? Wait, no, \( AB \) is 8, \( BC \) is 5, so \( AC = 8 + 5 = 13 \)? Wait, no, \( B \) is on \( AC \), so \( AC = AB + BC = 8 + 5 = 13 \)? Wait, no, the right angle is at \( B \), so \( \triangle ABD \) and \( \triangle CBD \) are right triangles, and \( \triangle ACD \) is also a right triangle? Wait, no, the diagram: \( A---B---C \) on a line, with \( B \) between \( A \) and \( C \), \( AB = 8 \), \( BC = 5 \), so \( AC = 13 \). Then \( D \) is a point such that \( DB \perp AC \), with \( DB = 15 \)? Wait, no, \( BD = 15 \), and we need to find \( AD = s \). Wait, no, in right triangle \( ABD \), right-angled at \( B \), so \( AD^2 = AB^2 + BD^2 \). Wait, \( AB = 8 \), \( BD = 15 \), so \( AD^2 = 8^2 + 15^2 = 64 + 225 = 289 \), so \( AD = \sqrt{289} = 17 \). Wait, but also, in right triangle \( CBD \), \( CD^2 = BC^2 + BD^2 = 5^2 + 15^2 = 25 + 225 = 250 \), but we need \( AD \), which is \( s \). Wait, maybe I misread the diagram. Wait, the problem is to find \( s \), which is \( AD \). So in right triangle \( ABD \), with legs \( AB = 8 \) and \( BD = 15 \), hypotenuse \( AD = s \). So by Pythagorean theorem:

\( s^2 = 8^2 + 15^2 \)

Step2: Calculate the squares

\( 8^2 = 64 \), \( 15^2 = 225 \)

Step3: Sum the squares

\( 64 + 225 = 289 \)

Step4: Take the square root

\( s = \sqrt{289} = 17 \)

Answer:

17