In triangle XYZ (not shown), the measure of `angleY` is `90^(@), YZ=12, and XZ=15.` Triangle HJK is similar to triangle XYZ, where vertices H, J, and K corresponds ot vertices X, Y, and Z, respectively, and each side of triangle HJK is `(1)/(5)` the length of the corresponding side of triangle XYZ. What is the value of tanK?

Answer

To solve the problem step by step, we will follow these instructions:

### Step 1: Identify the given information in triangle XYZ
- Angle Y = 90°
- YZ = 12 units (this is the base)
- XZ = 15 units (this is the hypotenuse)

### Step 2: Use the Pythagorean theorem to find XY
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (XZ) is equal to the sum of the squares of the other two sides (XY and YZ).

\[ XZ^2 = XY^2 + YZ^2 \]

Substituting the known values:

\[ 15^2 = XY^2 + 12^2 \]

Calculating the squares:

\[ 225 = XY^2 + 144 \]

Now, isolate XY^2:

\[ XY^2 = 225 - 144 \]
\[ XY^2 = 81 \]

Taking the square root to find XY:

\[ XY = \sqrt{81} = 9 \text{ units} \]

### Step 3: Identify the similar triangle HJK
Since triangle HJK is similar to triangle XYZ and each side of triangle HJK is \( \frac{1}{5} \) the length of the corresponding side of triangle XYZ, we can find the lengths of the sides of triangle HJK.

- HJ (corresponding to XY) = \( \frac{1}{5} \times 9 = \frac{9}{5} \) units
- JK (corresponding to YZ) = \( \frac{1}{5} \times 12 = \frac{12}{5} \) units
- HK (corresponding to XZ) = \( \frac{1}{5} \times 15 = 3 \) units

### Step 4: Find tan K
In triangle HJK, we need to find \( \tan K \). Since triangle HJK is similar to triangle XYZ, the angles are corresponding.

Using the definition of tangent:

\[ \tan K = \frac{\text{opposite}}{\text{adjacent}} = \frac{HJ}{JK} \]

Substituting the values we found:

\[ \tan K = \frac{\frac{9}{5}}{\frac{12}{5}} = \frac{9}{12} = \frac{3}{4} \]

### Conclusion
The value of \( \tan K \) is \( \frac{3}{4} \).

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