Question

question 12 of 25

what is the value of \\(x\\) in the diagram below?

a. 8
b. 7
c. 9
d. 6

Explanation:

Identify similar triangles

We observe two right triangles in the diagram. Both triangles have a right angle (\(90^\circ\)) and a marked acute angle that is congruent in both. By the Angle-Angle (AA) Similarity Postulate, these two triangles are similar.

Set up the proportion

Since the triangles are similar, the ratios of their corresponding sides are equal. We identify the corresponding parts:

• The base of the larger triangle is \(14\), and the base of the smaller triangle is \(2\).
• The hypotenuse of the larger triangle is \(49\), and the hypotenuse of the smaller triangle is \(x\).

We set up the ratio of corresponding sides:

$$\frac{\text{Hypotenuse of larger triangle}}{\text{Hypotenuse of smaller triangle}} = \frac{\text{Base of larger triangle}}{\text{Base of smaller triangle}}$$

Substituting the given values:

$$\frac{49}{x} = \frac{14}{2}$$

Solve for x

Simplify the fraction on the right side of the equation:

$$\frac{14}{2} = 7$$

Now, solve the simplified equation:

$$\frac{49}{x} = 7$$

$$7x = 49$$

$$x = 7$$

Answer:

• (A) 8
• (B) 7 (Correct answer)
• (C) 9
• (D) 6