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Triangle KLM
 is similar to triangle RTU
. The triangles are shown with side lengths in centimeters.
 

 
What is the length of side RU
 in centimeters?
A.2 1112
 cm
B.11 23
 cm 
C.6 12
 cm 
D.3 12
 cm 

The given triangles \( \triangle KLM \) and \( \triangle RTU \) are similar, meaning their corresponding sides are proportional. 

The side lengths of \( \triangle KLM \) are:
- \( MK = 14 \, \text{cm} \)
- \( KL = 10 \, \text{cm} \)
- \( ML = 12 \, \text{cm} \)

The side lengths of \( \triangle RTU \) are:
- \( TU = 2 \frac{1}{2} = 2.5 \, \text{cm} \)
- \( UT = 3 \, \text{cm} \)
- We are solving for \( RU \).

### Step-by-Step Approach:

We know that the triangles are similar, so the ratio of the sides should be equal:
\[
\frac{MK}{RU} = \frac{ML}{TU}
\]
Substituting the known values:
\[
\frac{14}{RU} = \frac{12}{3}
\]

Simplify the right-hand side:
\[
\frac{12}{3} = 4
\]

Now, solve for \( RU \):
\[
\frac{14}{RU} = 4
\]

Cross-multiply:
\[
14 = 4 \times RU
\]

Solve for \( RU \):
\[
RU = \frac{14}{4} = 3.5 \, \text{cm}
\]

Thus, the length of side \( RU \) is \( 3.5 \, \text{cm} \).

The correct answer is **D. 3 \frac{1}{2} \, \text{cm}**.

---

Would you like further clarification or details on any part of this process?

### Related Questions:
1. How do you calculate the missing side length in a triangle using proportions?
2. What does it mean when two triangles are similar?
3. How can we use the Pythagorean theorem with triangle dimensions?
4. How do we simplify fractional side lengths in similar triangles?
5. Can the area of similar triangles be compared using side ratios?

### Tip:
When working with similar triangles, always set up proportions carefully by matching the corresponding sides!

Triangle KLM is similar to triangle RTU. The triangles are shown with side lengths in centimeters. What is the length of side RU in centimeters? A. 2 11/12 cm B. 11 2/3 cm C. 6 1/2 cm D. 3 1/2 cm

This problem involves finding the missing side length in similar triangles KLM and RTU. Using proportions and cross-multiplication, the length of side RU is calculated as 3.5 cm. Detailed steps provided for Grades 7-10 students.

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