Question

consider δabc ~ δxyz. what is the value of tan(a)? what is the value of tan(x)? what is true about the two tangent ratios? (options for tan values: 3/4, 3/5, 4/5) images of right triangles δabc (legs 4, 3; hypotenuse 5) and δxyz (legs 8, 6; hypotenuse 10)

Explanation:

Step1: Recall the definition of tangent in a right triangle

In a right triangle, the tangent of an acute angle is the ratio of the length of the opposite side to the length of the adjacent side. For angle \( A \) in \( \triangle ABC \), the opposite side to \( A \) is \( BC \) and the adjacent side is \( AC \).

Step2: Identify the lengths of the opposite and adjacent sides for \( \angle A \)

From the diagram, \( BC = 3 \) and \( AC = 4 \).

Step3: Calculate \( \tan(A) \)

Using the tangent ratio formula \( \tan(\theta)=\frac{\text{opposite}}{\text{adjacent}} \), we substitute the values: \( \tan(A)=\frac{BC}{AC}=\frac{3}{4} \).

Answer:

\(\frac{3}{4}\)