Right triangle [tex]ABC[/tex] has side lengths [tex]AB = 6[/tex], [tex]BC = 8[/tex], and [tex]CA = 10[/tex]. A second right triangle has corresponding vertices of [tex]A^{\prime}, B^{\prime},[/tex] and [tex]C^{\prime}[/tex], with side lengths of 32, 40, and 24.

In 3-5 sentences, describe how to find the ratio of the side opposite [tex]\angle A[/tex] to the hypotenuse of triangle [tex]ABC[/tex]. Then use this ratio to identify the location of point [tex]A^{\prime}[/tex] in the second right triangle.

First, identify the side opposite [tex]\angle A[/tex] in triangle [tex]ABC[/tex], which is [tex]BC = 8[/tex]. The hypotenuse of triangle [tex]ABC[/tex] is [tex]CA = 10[/tex]. The ratio of the side opposite [tex]\angle A[/tex] to the hypotenuse is [tex]\frac{8}{10}[/tex] or [tex]0.8[/tex]. In the second triangle, this ratio helps us find that point [tex]A^{\prime}[/tex] is the vertex opposite the side measuring 32, because [tex]\frac{32}{40}[/tex] is also [tex]0.8[/tex].

Question

15-11-2023
Mathematics

College

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Right triangle [tex]ABC[/tex] has side lengths [tex]AB = 6[/tex], [tex]BC = 8[/tex], and [tex]CA = 10[/tex]. A second right triangle has corresponding vertices of [tex]A^{\prime}, B^{\prime},[/tex] and [tex]C^{\prime}[/tex], with side lengths of 32, 40, and 24.

In 3-5 sentences, describe how to find the ratio of the side opposite [tex]\angle A[/tex] to the hypotenuse of triangle [tex]ABC[/tex]. Then use this ratio to identify the location of point [tex]A^{\prime}[/tex] in the second right triangle.

First, identify the side opposite [tex]\angle A[/tex] in triangle [tex]ABC[/tex], which is [tex]BC = 8[/tex]. The hypotenuse of triangle [tex]ABC[/tex] is [tex]CA = 10[/tex]. The ratio of the side opposite [tex]\angle A[/tex] to the hypotenuse is [tex]\frac{8}{10}[/tex] or [tex]0.8[/tex]. In the second triangle, this ratio helps us find that point [tex]A^{\prime}[/tex] is the vertex opposite the side measuring 32, because [tex]\frac{32}{40}[/tex] is also [tex]0.8[/tex].

Answer :

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GinnyAnswer GinnyAnswer · Answer 1 · 2025-02-13T05:06:30-05:00

To find the ratio of the side opposite [tex]$\angle A$[/tex] to the hypotenuse in triangle [tex]$ABC$[/tex], first identify the relevant sides. In triangle [tex]$ABC$[/tex], side [tex]$BC = 8$[/tex] is opposite [tex]$\angle A$[/tex], and the hypotenuse is [tex]$CA = 10$[/tex]. The desired ratio is the length of [tex]$BC$[/tex] divided by the length of [tex]$CA$[/tex], which is [tex]$8/10$[/tex]. This simplifies to [tex]$0.8$[/tex].

For the second triangle [tex]$A'B'C'$[/tex], the triangles are similar, so the ratio of the side opposite [tex]$\angle A'$[/tex] to the hypotenuse should be the same, [tex]$0.8$[/tex]. Identifying the side lengths, we have side lengths 32, 40, and 24. If we take 24 as the hypotenuse, the side opposite [tex]$\angle A'$[/tex] should have the length [tex]$0.8 \times 24 = 19.2$[/tex]. This means the side opposite [tex]$\angle A'$[/tex] is 19.2 units long.