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To find the value of \(\tan M\), we need to understand the trigonometric function in a right-angled triangle. The tangent of an angle is defined as:

\[
\tan(\theta) = \frac{\text{opposite side}}{\text{adjacent side}}
\]

In the triangle provided, we need to find \(\tan M\). Here, \(M\) is the angle opposite side \(ON\) and adjacent to side \(ON\). From the triangle, we observe:

- Side \(ON = 15\) (adjacent to \(M\))
- Side \(OM = 25\) (hypotenuse, but not needed for tangent)
- Side \(MN\) is opposite to \(M\), but we need to find its length first.

### Step 1: Find the Length of \(MN\)
We can use the Pythagorean Theorem:

\[
OM^2 = ON^2 + MN^2
\]

\[
25^2 = 15^2 + MN^2
\]

\[
625 = 225 + MN^2
\]

\[
MN^2 = 625 - 225 = 400
\]

\[
MN = \sqrt{400} = 20
\]

### Step 2: Calculate \(\tan M\)

\[
\tan M = \frac{\text{opposite}}{\text{adjacent}} = \frac{MN}{ON} = \frac{20}{15}
\]

\[
\tan M = \frac{4}{3} \approx 1.33
\]

Therefore, the value of \(\tan M\), rounded to the nearest hundredth, is **1.33**.

Would you like further details or have any other questions?

### Related Questions:
1. How do you calculate sine or cosine for angle \(M\) in this triangle?
2. What is the hypotenuse of a right-angled triangle, and why is it not used for tangent?
3. Can you find \(\tan N\) in the same triangle?
4. What are the properties of the Pythagorean Theorem, and how is it applied here?
5. How can the trigonometric ratios be used to find missing angles in a triangle?

### Tip:
When working with right-angled triangles, always identify which sides are opposite, adjacent, and hypotenuse relative to the angle you are dealing with. This helps in correctly applying trigonometric ratios.

Find the value of tan M rounded to the nearest hundredth, if necessary, for the given right triangle with sides ON = 15 and OM = 25.

Learn how to calculate the tangent of angle M in a right triangle with sides ON = 15 and OM = 25. This problem involves the Pythagorean Theorem and trigonometric ratios, making it suitable for high school students. The solution is rounded to the nearest hundredth.

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