Question

find the unknown lengths in these pairs of overlapping triangles.
1.
2.

Explanation:

Step1: Identify similar - triangles

The two triangles in the first problem are similar. The ratio of corresponding sides of similar triangles is equal.

Step2: Set up the proportion

For the first pair of triangles, we have the proportion $\frac{7.5}{12}=\frac{5}{x}$.

Step3: Cross - multiply and solve for $x$

Cross - multiplying gives $7.5x = 12\times5$. Then $7.5x=60$, and $x=\frac{60}{7.5}=8$.

For the second pair of similar triangles, we set up the proportion $\frac{x}{x + 8}=\frac{15}{35}$.

Step4: Cross - multiply the second proportion

Cross - multiplying gives $35x=15(x + 8)$.

Step5: Expand and solve for $x$

Expand the right - hand side: $35x=15x+120$. Subtract $15x$ from both sides: $35x−15x=120$, so $20x = 120$, and $x = 6$.

Answer:

1. $x = 8$
2. $x = 6$