Answer :
Let's solve this problem step-by-step:
1. Identify the Known Information for Triangle ABC:
- In triangle [tex]\(ABC\)[/tex], we are given the side lengths:
- [tex]\(AB = 6\)[/tex]
- [tex]\(BC = 8\)[/tex]
- [tex]\(CA = 10\)[/tex] (which is the hypotenuse since [tex]\(ABC\)[/tex] is a right triangle)
2. Find the Ratio of the Side Opposite [tex]\(\angle A\)[/tex] to the Hypotenuse in Triangle ABC:
- In triangle [tex]\(ABC\)[/tex], the side [tex]\(BC\)[/tex] is opposite [tex]\(\angle A\)[/tex].
- We want to find the ratio of the side [tex]\(BC\)[/tex] (opposite [tex]\(\angle A\)[/tex]) to the hypotenuse [tex]\(CA\)[/tex].
- Ratio = [tex]\(\frac{BC}{CA} = \frac{8}{10} = 0.8\)[/tex]
3. Apply the Ratio to the Second Triangle [tex]\(A'B'C'\)[/tex]:
- For the second triangle [tex]\(A'B'C'\)[/tex], the side lengths given are:
- [tex]\(A'B' = 32\)[/tex]
- [tex]\(B'C' = 40\)[/tex]
- [tex]\(C'A' = 24\)[/tex] (we are told [tex]\(A'B'C'\)[/tex] corresponds to the sides of triangle [tex]\(ABC\)[/tex])
4. Identify the Location of Point [tex]\(A'\)[/tex]:
- Since we know [tex]\(C'A'\)[/tex] corresponds to the hypotenuse in triangle [tex]\(A'B'C'\)[/tex], and is opposite [tex]\(\angle A'\)[/tex], side [tex]\(C'\)[/tex] should have the corresponding measure.
- We will use the previously calculated ratio of [tex]\(0.8\)[/tex] to find the location of point [tex]\(A'\)[/tex].
- Using the same ratio in the context of triangle [tex]\(A'B'C'\)[/tex]:
- Side opposite [tex]\(\angle A'\)[/tex] = [tex]\(0.8 \times 24 = 19.2\)[/tex]
- Therefore, the point [tex]\(A'\)[/tex] is such that the side opposite to angle [tex]\(\angle A'\)[/tex] is 19.2.
This detailed explanation guides you through finding the ratio of the triangle [tex]\(ABC\)[/tex] and then using it to identify the location of point [tex]\(A'\)[/tex] in the triangle [tex]\(A'B'C'\)[/tex].
1. Identify the Known Information for Triangle ABC:
- In triangle [tex]\(ABC\)[/tex], we are given the side lengths:
- [tex]\(AB = 6\)[/tex]
- [tex]\(BC = 8\)[/tex]
- [tex]\(CA = 10\)[/tex] (which is the hypotenuse since [tex]\(ABC\)[/tex] is a right triangle)
2. Find the Ratio of the Side Opposite [tex]\(\angle A\)[/tex] to the Hypotenuse in Triangle ABC:
- In triangle [tex]\(ABC\)[/tex], the side [tex]\(BC\)[/tex] is opposite [tex]\(\angle A\)[/tex].
- We want to find the ratio of the side [tex]\(BC\)[/tex] (opposite [tex]\(\angle A\)[/tex]) to the hypotenuse [tex]\(CA\)[/tex].
- Ratio = [tex]\(\frac{BC}{CA} = \frac{8}{10} = 0.8\)[/tex]
3. Apply the Ratio to the Second Triangle [tex]\(A'B'C'\)[/tex]:
- For the second triangle [tex]\(A'B'C'\)[/tex], the side lengths given are:
- [tex]\(A'B' = 32\)[/tex]
- [tex]\(B'C' = 40\)[/tex]
- [tex]\(C'A' = 24\)[/tex] (we are told [tex]\(A'B'C'\)[/tex] corresponds to the sides of triangle [tex]\(ABC\)[/tex])
4. Identify the Location of Point [tex]\(A'\)[/tex]:
- Since we know [tex]\(C'A'\)[/tex] corresponds to the hypotenuse in triangle [tex]\(A'B'C'\)[/tex], and is opposite [tex]\(\angle A'\)[/tex], side [tex]\(C'\)[/tex] should have the corresponding measure.
- We will use the previously calculated ratio of [tex]\(0.8\)[/tex] to find the location of point [tex]\(A'\)[/tex].
- Using the same ratio in the context of triangle [tex]\(A'B'C'\)[/tex]:
- Side opposite [tex]\(\angle A'\)[/tex] = [tex]\(0.8 \times 24 = 19.2\)[/tex]
- Therefore, the point [tex]\(A'\)[/tex] is such that the side opposite to angle [tex]\(\angle A'\)[/tex] is 19.2.
This detailed explanation guides you through finding the ratio of the triangle [tex]\(ABC\)[/tex] and then using it to identify the location of point [tex]\(A'\)[/tex] in the triangle [tex]\(A'B'C'\)[/tex].
