Right triangle ABC has side lengths AB=6 , BC=8 , and CA=10 . A second right triangle has corresponding vertices of A′ , B′ , and C′ , with side lengths of 32, 40, and 24. In 3–5 sentences, describe how to find the ratio of the side opposite ∠A to the hypotenuse of triangle ABC . Then use this ratio to identify the location of point A′ in the second right triangle.

Question

Right triangle  ABC  has side lengths  AB=6 ,  BC=8 , and CA=10 . A second right triangle has corresponding vertices of A′ , B′ , and C′ , with side lengths of 32, 40, and 24. In 3–5 sentences, describe how to find the ratio of the side opposite  ∠A  to the hypotenuse of triangle  ABC . Then use this ratio to identify the location of point  A′  in the second right triangle.

GPT-4o mini · Answer

To find the ratio of the side opposite ∠A to the hypotenuse of triangle ABC, we first identify the lengths of the sides: AB (the side opposite ∠A) is 6, and AC (the hypotenuse) is 10. The ratio can then be calculated as follows: ratio = AB / AC = 6 / 10 = 3/5.

Now, to find the corresponding point A′ in triangle A′B′C′, we apply this same ratio to the second triangle. The hypotenuse of triangle A′B′C′ is 40, so we calculate the side opposite ∠A′ (which corresponds to A) as: A′B′ / A′C′ = 3/5. Thus, the length of A′B′ can be determined as (3/5) * 40 = 24. Therefore, point A′ is located 24 units from vertex B′ along the longer leg of the right triangle A′B′C′.