ON is x , MN is 10, OM is x+2, O is a right angle. Find angle M, N, and side o = 10, find side m and n

Let's break down the problem to solve for the angles \( M \), \( N \), and the sides \( m \) and \( n \) of the right triangle \( \triangle OMN \), where \( O \) is the right angle.

### Given:
- \( \angle O = 90^\circ \)
- \( ON = x \)
- \( OM = x + 2 \)
- \( MN = 10 \) (This is the hypotenuse, let's denote it as \( c \))
- \( ON = m \) and \( OM = n \) (sides \( m \) and \( n \) need to be found)

### Step 1: Use the Pythagorean Theorem
For any right triangle, the Pythagorean theorem states:
\[
m^2 + n^2 = c^2
\]

Substituting the given values:
\[
x^2 + (x + 2)^2 = 10^2
\]

Expanding and simplifying:
\[
x^2 + (x^2 + 4x + 4) = 100
\]
\[
2x^2 + 4x + 4 = 100
\]
\[
2x^2 + 4x - 96 = 0
\]
\[
x^2 + 2x - 48 = 0
\]

### Step 2: Solve the Quadratic Equation
Solve the quadratic equation using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Where \( a = 1 \), \( b = 2 \), and \( c = -48 \):
\[
x = \frac{-2 \pm \sqrt{2^2 + 4 \times 48}}{2 \times 1}
\]
\[
x = \frac{-2 \pm \sqrt{4 + 192}}{2}
\]
\[
x = \frac{-2 \pm \sqrt{196}}{2}
\]
\[
x = \frac{-2 \pm 14}{2}
\]

This gives two possible solutions:
\[
x = \frac{12}{2} = 6 \quad \text{or} \quad x = \frac{-16}{2} = -8
\]

Since \( x \) represents a length, it must be positive, so \( x = 6 \).

### Step 3: Find the Lengths of \( m \) and \( n \)
Now that we have \( x = 6 \):
\[
ON = m = 6, \quad OM = n = x + 2 = 6 + 2 = 8
\]

### Step 4: Find the Angles \( \angle M \) and \( \angle N \)
Use trigonometric ratios to find the angles.

For \( \angle M \):
\[
\sin M = \frac{ON}{MN} = \frac{6}{10} = 0.6
\]
\[
M = \arcsin(0.6) \approx 36.87^\circ
\]

For \( \angle N \):
Since the angles in a triangle sum to \( 180^\circ \):
\[
N = 90^\circ - M = 90^\circ - 36.87^\circ = 53.13^\circ
\]

### Final Answers:
- \( \angle M \approx 36.87^\circ \)
- \( \angle N \approx 53.13^\circ \)
- \( m = 6 \)
- \( n = 8 \)
- \( c = 10 \)

Would you like more details or have any questions? 

### 8 Related Questions:
1. What are the basic trigonometric ratios in a right triangle?
2. How do you derive the sine, cosine, and tangent values for a right triangle?
3. What is the relationship between the angles in a right triangle?
4. How do you solve a quadratic equation?
5. What is the importance of the Pythagorean theorem in geometry?
6. How can you find the area of a right triangle using its sides?
7. What are the properties of a right triangle?
8. How do you calculate the angles of a right triangle if two sides are known?

### Tip:
When solving right triangles, always remember that the sum of the angles in a triangle is \(180^\circ\), and the Pythagorean theorem is your go-to tool for relating the sides.

Learn how to solve a right triangle problem to find angles \( M \) and \( N \), and sides \( m \) and \( n \). Detailed step-by-step solution using trigonometric ratios and the Pythagorean theorem. Suitable for Grades 10-12.

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