To find the lengths of the legs of a right triangle where an altitude to the hypotenuse divides the hypotenuse into segments of 9 cm and 16 cm, follow these steps:
### Step 1: Geometric Mean Theorem
First, use the geometric mean theorem. This theorem states that the length of the altitude (h) to the hypotenuse of a right triangle is the geometric mean of the lengths of the segments it divides the hypotenuse into.
Given segments:
- Segment [tex]\(a = 9 \text{ cm}\)[/tex]
- Segment [tex]\(b = 16 \text{ cm}\)[/tex]
The altitude [tex]\(h\)[/tex] is calculated by:
[tex]\[ h = \sqrt{a \cdot b} = \sqrt{9 \cdot 16} \][/tex]
### Step 2: Simplifying the Altitude Calculation
[tex]\[ h = \sqrt{144} \][/tex]
[tex]\[ h = 12 \text{ cm} \][/tex]
So, the altitude is 12 cm.
### Step 3: Use Pythagorean Theorem for Triangle Legs
Next, we need to find the lengths of the legs of the right triangle. These legs can be found using the properties of similar triangles and the Pythagorean Theorem.
The lengths of the legs (let's call them [tex]\( leg_1 \)[/tex] and [tex]\( leg_2 \)[/tex]) are connected with the segments created by the altitude. Given the original segments of the hypotenuse:
- [tex]\( leg_1 \)[/tex] corresponds to the segment of the hypotenuse that is divided into 9 cm.
- [tex]\( leg_2 \)[/tex] corresponds to the segment of the hypotenuse that is divided into 16 cm.
We calculate the lengths of these legs by taking the square root of each segment multiplied by the sum of the segments and then divided by two.
For [tex]\( leg_1 \)[/tex]:
[tex]\[ \text{leg}_1 = \sqrt{a \cdot (a + b)} \][/tex]
[tex]\[ \text{leg}_1 = \sqrt{9 \cdot (9 + 16)} \][/tex]
### Step 4: Final Calculations
[tex]\[ \text{leg}_1 = \sqrt{9 \cdot 25} \][/tex]
[tex]\[ \text{leg}_1 = \sqrt{225} \][/tex]
[tex]\[ \text{leg}_1 = 15 \text{ cm} \][/tex]
For [tex]\( leg_2 \)[/tex]:
[tex]\[ \text{leg}_2 = \sqrt{b \cdot (a + b)} \][/tex]
[tex]\[ \text{leg}_2 = \sqrt{16 \cdot (9 + 16)} \][/tex]
[tex]\[ \text{leg}_2 = \sqrt{16 \cdot 25} \][/tex]
[tex]\[ \text{leg}_2 = \sqrt{400} \][/tex]
[tex]\[ \text{leg}_2 = 20 \text{ cm} \][/tex]
### Conclusion
The lengths of the legs of the right triangle are:
- [tex]\( \text{leg}_1 = 15 \text{ cm} \)[/tex]
- [tex]\( \text{leg}_2 = 20 \text{ cm} \)[/tex]
Therefore, the lengths of the legs of this triangle are 15 cm and 20 cm.
