Question

what are two different ways you could find the value of a? explain these methods.

Explanation:

Method 1: Geometric Mean (Leg) Theorem

The Geometric Mean (Leg) Theorem states that in a right triangle, the length of a leg (in this case, \(a\)) is the geometric mean of the length of the hypotenuse segment adjacent to that leg (16) and the length of the entire hypotenuse. First, we find the entire hypotenuse: the two segments are 9 and 16, so the hypotenuse \(c = 9 + 16 = 25\). Then, by the Geometric Mean (Leg) Theorem, \(a=\sqrt{16\times25}\). We calculate \(16\times25 = 400\), and \(\sqrt{400}=20\), so \(a = 20\).

Method 2: Pythagorean Theorem (Two - Step)

1. First, find the height (let's call it \(h\)) of the right triangle using the left - hand right triangle. The left - hand right triangle has a hypotenuse of 15 and one leg of 9. By the Pythagorean Theorem \(h=\sqrt{15^{2}-9^{2}}\). Calculate \(15^{2}=225\) and \(9^{2}=81\), so \(h=\sqrt{225 - 81}=\sqrt{144}=12\).
2. Then, use the right - hand right triangle (with leg \(h = 12\) and one leg of 16) to find \(a\) by the Pythagorean Theorem. So \(a=\sqrt{12^{2}+16^{2}}\). Calculate \(12^{2}=144\) and \(16^{2}=256\), then \(144 + 256=400\), and \(\sqrt{400}=20\), so \(a = 20\).

Answer:

Two methods to find \(a\) are:

1. Geometric Mean (Leg) Theorem: The hypotenuse of the large right triangle is \(9 + 16=25\). By the Geometric Mean (Leg) Theorem, \(a=\sqrt{16\times25}=\sqrt{400} = 20\).
2. Pythagorean Theorem (Two - Step):

• Step 1: Find the height \(h\) of the large right triangle using the left - hand right triangle with hypotenuse 15 and leg 9. \(h=\sqrt{15^{2}-9^{2}}=\sqrt{225 - 81}=\sqrt{144}=12\).
• Step 2: Find \(a\) using the right - hand right triangle with legs \(h = 12\) and 16. \(a=\sqrt{12^{2}+16^{2}}=\sqrt{144 + 256}=\sqrt{400}=20\).