Question

finding side lengths in a right triangle
what is the value of s?
diagram of right triangle with points a, b, c, d; ab=8, bc=5, bd=15, ad=s, right angle at b
□ units

Explanation:

Step1: Identify Right Triangles

We have two right triangles: \( \triangle ABD \) (right-angled at \( B \)) and \( \triangle CBD \) (right-angled at \( B \)). By the Geometric Mean (Leg) Theorem, in a right triangle, the length of a leg is the geometric mean of the lengths of the hypotenuse segments. Also, we can use the Pythagorean theorem in \( \triangle ABD \) and \( \triangle CBD \), but notice that \( AD = s \), \( AB = 8 \), \( BD = 15 \), and we can also consider the hypotenuse \( AC = AB + BC = 8 + 5 = 13 \)? Wait, no, \( AB = 8 \), \( BC = 5 \), so \( AC = 8 + 5 = 13 \)? Wait, no, actually, in right triangles \( \triangle ABD \) and \( \triangle CBD \), we can use the Pythagorean theorem for \( \triangle ABD \): \( AD^2 = AB^2 + BD^2 \)? Wait, no, \( \triangle ABD \) is right-angled at \( B \), so \( AD^2 = AB^2 + BD^2 \). Wait, \( AB = 8 \), \( BD = 15 \), so \( AD^2 = 8^2 + 15^2 \).

Step2: Calculate \( AD^2 \)

\( 8^2 = 64 \), \( 15^2 = 225 \), so \( AD^2 = 64 + 225 = 289 \).

Step3: Find \( AD \) (which is \( s \))

Take the square root of 289: \( \sqrt{289} = 17 \). So \( s = 17 \).

Answer:

17