Question

right triangle similarity theorem

1. the corresponding sides of the similar triangles

original triangle\tnew larger triangle\tnew smaller triangle
hypotenuse
longer leg
shorter leg
solve for the geometric means a, b, and s.
geometric means\tproportion\tanswer
altitude a
shorter leg s
longer leg b

Explanation:

Step1: Find Altitude \( a \)

In a right triangle, the altitude to the hypotenuse is the geometric mean of the segments into which it divides the hypotenuse. The segments are 2 and 8. So the proportion is \( \frac{2}{a}=\frac{a}{8} \), which gives \( a^2 = 2\times8 = 16 \), so \( a = \sqrt{16}=4 \).

Step2: Find Shorter Leg \( s \)

The shorter leg of the original triangle is the geometric mean of the hypotenuse segment adjacent to it (2) and the entire hypotenuse (2 + 8 = 10? Wait, no, wait. Wait, the segments are 2 and 8, so the hypotenuse is 2 + 8 = 10? Wait, no, actually, the shorter leg \( s \) (let's say the side with length \( s \)) is the geometric mean of the segment (2) and the hypotenuse (2 + 8 = 10? Wait, no, the formula is: in a right triangle, each leg is the geometric mean of the hypotenuse and the segment of the hypotenuse adjacent to that leg. Wait, the shorter leg (let's say the one with length \( s \)) is adjacent to the segment of length 2, and the longer leg \( b \) is adjacent to the segment of length 8. The hypotenuse is 2 + 8 = 10? Wait, no, looking at the diagram, the segments are 2 and 8, so the hypotenuse \( h = 2 + 8 = 10 \)? Wait, no, the diagram shows two segments of length 2 (maybe a typo? Wait, the diagram has a segment of length 2 and 8, and the altitude \( a \). Wait, maybe the hypotenuse is split into 2 and 8, so the hypotenuse is 2 + 8 = 10. Then the shorter leg \( s \) (the leg opposite the smaller angle) is the geometric mean of the segment (2) and the hypotenuse (10)? No, that's not right. Wait, the correct formula is: if a right triangle is divided into two smaller similar triangles by the altitude to the hypotenuse, then each leg is the geometric mean of the hypotenuse and the segment of the hypotenuse adjacent to that leg. So, let's denote:

- Let the original right triangle have hypotenuse \( h = 2 + 8 = 10 \) (assuming the two segments are 2 and 8).

- Let the shorter leg be \( s \) (adjacent to the segment of length 2), so \( s^2 = 2\times h = 2\times10 = 20 \)? No, that's not matching. Wait, maybe the segments are 2 and 8, and the altitude is \( a \), and the legs are \( s \) (shorter) and \( b \) (longer), and the hypotenuse is \( h \). Wait, maybe the diagram has a segment of length 2 (the smaller segment) and 8 (the larger segment), so the hypotenuse is \( 2 + 8 = 10 \). Then:

- Altitude \( a \): \( a^2 = 2\times8 = 16 \), so \( a = 4 \) (as we found earlier).

- Shorter leg \( s \): \( s^2 = 2\times(2 + 8) = 2\times10 = 20 \)? No, that's not. Wait, no, the correct formula is: in the right triangle, the leg (shorter) is the geometric mean of the segment adjacent to it (2) and the hypotenuse (2 + 8 = 10). Wait, no, the correct formula is \( \text{leg}^2 = \text{segment} \times \text{hypotenuse} \). So for the shorter leg (adjacent to segment 2), \( s^2 = 2\times(2 + 8) = 2\times10 = 20 \), so \( s = \sqrt{20} = 2\sqrt{5} \)? No, that can't be. Wait, maybe the diagram has the two segments as 2 and 8, and the altitude \( a \), and the legs are \( s \) (length \( s \)) and \( b \) (length \( b \)), and the hypotenuse is \( h \). Wait, maybe the problem is that the two segments are 2 and 8, so the hypotenuse is 2 + 8 = 10, and the altitude is \( a = 4 \) (from step 1). Then the shorter leg \( s \) (the leg with length \( s \)) is the geometric mean of the segment (2) and the hypotenuse (10)? No, that's not. Wait, the correct formula is: each leg is the geometric mean of the hypotenuse and the segment of the hypotenuse adjacent to that leg. So:

- Shorter leg \( s \): adjacent to segment 2, so \( s^2 = 2\times h \), where \( h = 2 + 8 = 10 \), so \( s^2 = 2\times10 = 20 \), \( s = \sqrt{20} = 2\sqrt{5} \approx 4.47 \). But that doesn't seem right. Wait, maybe the segments are 2 and 8, and the hypotenuse is 2 + 8 = 10, and the legs are \( s \) and \( b \), and the altitude is \( a = 4 \). Then the longer leg \( b \) is the geometric mean of the segment 8 and the hypotenuse 10, so \( b^2 = 8\times10 = 80 \), \( b = \sqrt{80} = 4\sqrt{5} \approx 8.94 \). But let's check with the altitude: in a right triangle, \( a^2 + s^2 = (2)^2 + a^2 \)? No, wait, the area of the triangle can be calculated as \( \frac{1}{2} \times h \times a = \frac{1}{2} \times 10 \times 4 = 20 \), and also as \( \frac{1}{2} \times s \times b \). If \( s = 2\sqrt{5} \) and \( b = 4\sqrt{5} \), then \( \frac{1}{2} \times 2\sqrt{5} \times 4\sqrt{5} = \frac{1}{2} \times 40 = 20 \), which matches. So that works.

Wait, but maybe the diagram has the two segments as 2 and 8, so the hypotenuse is 2 + 8 = 10, altitude \( a = 4 \), shorter leg \( s = \sqrt{2\times10} = \sqrt{20} = 2\sqrt{5} \), longer leg \( b = \sqrt{8\times10} = \sqrt{80} = 4\sqrt{5} \).

But let's re-examine the problem. The problem says "Solve for the geometric means a, b, and s".

For Altitude \( a \):

The altitude to the hypotenuse is the geometric mean of the two segments, so \( a = \sqrt{2\times8} = \sqrt{16} = 4 \). That's correct.

For Shorter Leg \( s \):

The shorter leg is the geometric mean of the segment adjacent to it (2) and the hypotenuse (2 + 8 = 10). Wait, no, the correct formula is: in a right triangle, when you draw the altitude to the hypotenuse, each leg is the geometric mean of the hypotenuse and the segment of the hypotenuse adjacent to that leg. So, if the hypotenuse is \( h = p + q \) (where \( p = 2 \), \( q = 8 \)), then the leg adjacent to \( p \) (shorter leg \( s \)) satisfies \( s^2 = p \times h = 2 \times (2 + 8) = 2 \times 10 = 20 \), so \( s = \sqrt{20} = 2\sqrt{5} \approx 4.47 \).

For Longer Leg \( b \):

The longer leg \( b \) is adjacent to the segment \( q = 8 \), so \( b^2 = q \times h = 8 \times 10 = 80 \), so \( b = \sqrt{80} = 4\sqrt{5} \approx 8.94 \).

But let's check with the Pythagorean theorem: \( s^2 + b^2 = (2\sqrt{5})^2 + (4\sqrt{5})^2 = 20 + 80 = 100 = 10^2 = h^2 \), which is correct. And \( a^2 + 2^2 = 4^2 + 2^2 = 16 + 4 = 20 = s^2 \), which is correct (since the smaller triangle with legs \( a \) and 2 has hypotenuse \( s \)). Similarly, \( a^2 + 8^2 = 16 + 64 = 80 = b^2 \), which is correct (the larger triangle with legs \( a \) and 8 has hypotenuse \( b \)).

So summarizing:

- Altitude \( a \): \( \sqrt{2\times8} = 4 \)

- Shorter Leg \( s \): \( \sqrt{2\times(2 + 8)} = \sqrt{20} = 2\sqrt{5} \) (or approximately 4.47)

- Longer Leg \( b \): \( \sqrt{8\times(2 + 8)} = \sqrt{80} = 4\sqrt{5} \) (or approximately 8.94)

But maybe the diagram has the hypotenuse split into 2 and 8, so the hypotenuse is 2 + 8 = 10, and the legs are \( s \) (shorter) and \( b \) (longer), altitude \( a = 4 \).

Answer:

• Altitude \( a \): \( 4 \)
• Shorter Leg \( s \): \( 2\sqrt{5} \) (or \( \approx 4.47 \))
• Longer Leg \( b \): \( 4\sqrt{5} \) (or \( \approx 8.94 \))