https://file.aiquickdraw.com/mathbot/file/1726617456048ftzyal1m.jpg

In the given diagram, we see two right-angled triangles, \( \triangle MNZ \) and \( \triangle MNP \). Let's solve for \( x \) by using the properties of similar triangles or trigonometry.

### Step 1: Identifying Similar Triangles
- \( \triangle MNZ \) and \( \triangle MNP \) are similar because both are right-angled triangles that share the angle \( \angle M \).
  
From similar triangles, the ratios of corresponding sides are equal:
\[
\frac{MN}{MP} = \frac{NZ}{NP} = \frac{MZ}{MN}
\]

### Step 2: Using the Corresponding Side Ratios
Let's set up the equation using the corresponding sides of the similar triangles. In \( \triangle MNZ \), the sides are \( MN = 4 \, \text{cm} \), \( MZ = x \, \text{cm} \), and in \( \triangle MNP \), the sides are \( MP = 12 \, \text{cm} \) and \( NP = 4 + NZ \, \text{cm} \).

Using the fact that \( \frac{MN}{MP} = \frac{NZ}{NP} \), we can proceed to solve for \( x \).

Let me compute this next.The value of \( x \), which is the length of \( MZ \), is approximately \( 1.33 \, \text{cm} \).

Would you like more details or have any questions?

Here are five related questions for further practice:
1. How would you calculate the height of a right triangle given two sides?
2. Can you find the hypotenuse of triangle \( MNP \) using the Pythagorean theorem?
3. How can you determine whether two triangles are similar?
4. What other methods can be used to find side lengths in right triangles?
5. How would the solution change if the triangle had different side lengths?

**Tip**: In similar triangles, corresponding sides have proportional lengths.

Solve for x in the given right-angled triangle with sides 4 cm, x cm, and 12 cm.

This problem involves solving for the unknown side x in a right-angled triangle using the concept of similar triangles. The side lengths of the triangle are 4 cm and 12 cm. Detailed steps are provided to solve the problem using the proportionality of sides in similar triangles.

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