Given a right triangle $△abc$ where angle $a$ is one of the acute angles, if $\sin \left(a\right)=\frac{3}{5}$, $\cos \left(a\right)=\frac{4}{5}$, and $\tan \left(a\right)=\frac{3}{4}$, which of the following triangles is similar to $△abc$?

• A. A right triangle with side lengths $5$, $12$, and $13$
• B. A right triangle with side lengths $8$, $15$, and $17$
• C. A right triangle with side lengths $9$, $12$, and $15$
• D. A right triangle with side lengths $6$, $8$, and $10$

Answer: (D)

Verified step by step guidance

1. Identify the given trigonometric ratios for angle \( a \): \( \sin(a) = \frac{3}{5} \), \( \cos(a) = \frac{4}{5} \), and \( \tan(a) = \frac{3}{4} \). These ratios correspond to the sides of a right triangle where the opposite side to angle \( a \) is 3 units, the adjacent side is 4 units, and the hypotenuse is 5 units.
2. Recognize that the triangle with sides 3, 4, and 5 is a right triangle and serves as a reference for similarity. Any triangle similar to this one must have side lengths proportional to 3, 4, and 5.
3. For each given triangle option, check if the side lengths are in the same ratio as 3:4:5 by dividing each side by the smallest side and comparing the ratios.
4. If the ratios of the sides in a given triangle match the ratios 3:4:5 (or a constant multiple thereof), then that triangle is similar to the original triangle \( \triangle abc \).
5. Conclude that the triangle with side lengths 6, 8, and 10 is similar to \( \triangle abc \) because its sides are exactly twice the lengths of 3, 4, and 5, maintaining the same ratio and thus the same angles.