To find the missing sides of triangles, we can use the properties of similar triangles, which include proportionality of corresponding sides.

Problem 1: $\triangle LKJ \sim \triangle FED$

Given:

• $LJ = 72$, $FE = 32$
• Missing side $LK$ corresponds to $FD$

The proportion of corresponding sides is: $\frac{LK}{FD} = \frac{LJ}{FE} = \frac{72}{32}$ Simplifying this ratio: $\frac{72}{32} = \frac{9}{4}$. So the missing side $LK$ is: $LK = 72 \times \frac{9}{4} = 162$ Missing side $LK = 162$.

Problem 2: $\triangle QRS \sim \triangle KL$

Given:

• $QR = 36$, $KL = 11$
• Missing side $RS$

We use the ratio of corresponding sides: $\frac{QR}{KL} = \frac{RS}{LK}$ The ratio is $\frac{36}{11}$. So the missing side $RS$ is: $RS = LK \times \frac{36}{11} = 33 \times \frac{36}{11} = 108$ Missing side $RS = 108$.

Problem 3: $\triangle LMN \sim \triangle GFE$

Given:

• $LN = 10$, $GE = 30$
• Missing side $LM$

Use the ratio: $\frac{LM}{GF} = \frac{LN}{GE} = \frac{10}{30} = \frac{1}{3}$ Thus, the missing side is: $LM = GF \times \frac{1}{3} = 21 \times \frac{1}{3} = 7$ Missing side $LM = 7$.

Would you like further details on these or other problems?

Here are 5 related questions:

1. How are similar triangles used in solving real-world problems?
2. What are the properties of similar triangles?
3. How can we use the Pythagorean theorem alongside similarity?
4. What types of transformations preserve triangle similarity?
5. How do dilation and scaling factors relate to triangle similarity?

Tip: Always check if two triangles are similar by comparing their angles or using proportional sides!