Question

A right triangle ABC has side lengths AB=6 BC=8 and CA=10. a second right triangle has corresponding vertices of A', B', C', with side lengths of 32, 40, 24. 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.

Answer 1

To find the ratio of the side opposite angle \( A \) to the hypotenuse of triangle \( ABC \), we first identify the lengths of the sides of triangle \( ABC \). In triangle \( ABC \):

• Side \( AB = 6 \) (this is one leg),
• Side \( BC = 8 \) (this is the other leg),
• Side \( CA = 10 \) (this is the hypotenuse).

The angle \( A \) is opposite side \( BC \). Therefore, the side opposite angle \( A \) is the length of side \( BC\), which is 8.

Now, to find the ratio of the side opposite angle \( A \) to the hypotenuse, we can express it as follows:

\[ \text{Ratio} = \frac{\text{Opposite side}}{\text{Hypotenuse}} = \frac{BC}{CA} = \frac{8}{10} = \frac{4}{5}. \]

This ratio \( \frac{4}{5} \) indicates the relationship between the opposite side to angle \( A \) and the hypotenuse in triangle \( ABC \).

Now we will identify the corresponding side lengths in triangle \( A'B'C' \). In triangle \( A'B'C' \), the corresponding lengths of the sides are:

• Opposite side \( A' B' = 32\),
• Opposite side \( A' C' = 40\) (the hypotenuse),
• Side \( B' C' = 24\).

To find the corresponding location of point \( A' \) in triangle \( A'B'C' \) using the ratio \( \frac{4}{5}\):

1. Let \( x \) be the length of the side opposite angle \( A' \) in triangle \( A'B'C' \). We need to find \( A'B' \) in terms of the hypotenuse \( A' C' \).
2. The ratio of the side opposite angle \( A' \) (which corresponds to side \( BC \) in triangle \( ABC \)) to the hypotenuse \( A'C' \) (which corresponds to side \( CA \) in triangle \( ABC \)) gives us:

\[ \frac{A'B'}{A'C'} = \frac{4}{5}. \]

3. Given \( A'C' = 40 \), we can find \( A'B' \):

\[ A'B' = \frac{4}{5} \times A'C' = \frac{4}{5} \times 40 = 32. \]

Thus, the side opposite to angle \( A' \) in triangle \( A'B'C' \) (denoted as \( A'B' \)) has been confirmed to be \( 32 \), which indicates that point \( A' \) is located where the side opposite angle \( A' \) and the hypotenuse \( A'C' \) maintains the same ratio \( \frac{4}{5} \).

In summary, point \( A' \) is located based on the ratio \( \frac{BC}{CA} = \frac{4}{5} \) which aligns with the sides \( 32 \) (the opposite side) and \( 40 \) (the hypotenuse) of triangle \( A'B'C' \).

Answer 2

A. the ratio of the opposite side to the hypotenuse is 4/3, and point is opposite the side that has length 40.

B. the ratio of the opposite side to the hypotenuse is 4/5, and point is opposite the side that has length 32.

C. the ratio of the opposite side to the hypotenuse is 3/5, and point is opposite the side that has length 24.

Answer 3

Let's clarify the triangles and their relationships based on the information provided:

Given:

1. Triangle ABC (Right triangle):

   • \( AB = 6 \) (one leg)

   • \( BC = 8 \) (the other leg, opposite angle A)

   • \( CA = 10 \) (the hypotenuse)

   • So, we have:

     • Side opposite angle A = BC = 8
     • Hypotenuse = CA = 10

2. Triangle A'B'C' (Another right triangle):

   • Corresponding sides:
     • Length opposite angle A' = \( B'C' = 24 \)
     • \( A'B' = 32 \)
     • Hypotenuse \( A'C' = 40 \)

Now, let's evaluate the options:

A. The ratio of the opposite side to the hypotenuse is \( \frac{4}{3} \), and point is opposite the side that has length 40.

• Incorrect: The ratio of the side opposite \( A \) (8) to the hypotenuse (10) in triangle ABC is \( \frac{8}{10} = \frac{4}{5} \). The side corresponding to the hypotenuse (A'C' = 40) in triangle A'B'C' indicates that the opposite angle A' relates to side \( B'C' \).

B. The ratio of the opposite side to the hypotenuse is \( \frac{4}{5} \), and point is opposite the side that has length 32.

• Correct: The ratio \( \frac{4}{5} \) is correct (as shown above), with \( A'B' = 32 \) being the side adjacent to angle A' (opposite the opposite side \( B'C' = 24 \)).

C. The ratio of the opposite side to the hypotenuse is \( \frac{3}{5} \), and point is opposite the side that has length 24.

• Incorrect: The ratio is not \( \frac{3}{5} \). The side \( B'C' \), which is opposite angle A', is indeed \( 24 \). However, the ratios do not match.

Conclusion:

The correct option is:

B. The ratio of the opposite side to the hypotenuse is \( \frac{4}{5} \), and point \( A' \) is opposite the side that has length \( 32 \).

In triangle \( A'B'C' \), the side opposite angle \( A' \) is \( B'C' = 24 \). So, although point \( A' \) is indeed located opposite the side that corresponds to the length of \( 40 \) (the hypotenuse), the ratio pertaining to triangle \( ABC \ remains ( \frac{8}{10} = \frac{4}{5} \), confirming option B as accurate in both its ratio and side association.

Answer 4

u sure its correct

Answer 5

Let's clarify the choices once again to ensure we have the correct assessment based on triangle properties and relationships.

Given Information Recap

Triangle ABC:

• \( AB = 6 \) (one leg)
• \( BC = 8 \) (the other leg, opposite angle \( A \))
• \( CA = 10 \) (the hypotenuse)

So we can establish:

• \( \text{Opposite side (to } A \text{)} = BC = 8 \)
• \( \text{Hypotenuse} = CA = 10 \)

The ratio of the side opposite angle \( A \) to the hypotenuse is: \[ \text{Ratio} = \frac{BC}{CA} = \frac{8}{10} = \frac{4}{5}. \]

Triangle A'B'C':

• Side lengths:
  • \( A'B' = 32 \)
  • \( B'C' = 24 \) (opposite angle \( A' \))
  • \( A'C' = 40 \) (the hypotenuse)

Evaluating Choices Again

• Choice A:

  • "The ratio of the opposite side to the hypotenuse is \( \frac{4}{3} \), and point is opposite the side that has length 40."
  • Incorrect: The ratio is \( \frac{4}{5} \), not \( \frac{4}{3} \).

• Choice B:

  • "The ratio of the opposite side to the hypotenuse is \( \frac{4}{5} \), and point is opposite the side that has length 32."
  • Partially Correct: The ratio \( \frac{4}{5} \) is indeed correct. However, point \( A' \) is opposite the side \( B'C' = 24\), while side \( A'B' = 32 \) is adjacent to angle \( A' \).

• Choice C:

  • "The ratio of the opposite side to the hypotenuse is \( \frac{3}{5} \), and point is opposite the side that has length 24."
  • Incorrect: The ratio is not \( \frac{3}{5} \). However, \( B'C' = 24 \) is opposite angle \( A' \) which corresponds to the correct placement of point \( A' \).

Clarification of Identifying Points

Given the side relationships in triangle \( A'B'C' \):

• The side \( B'C' = 24\) corresponds to the opposite side to angle \( A'\).
• The hypotenuse \( A'C' = 40 \).

Conclusion

• The correct ratio of the opposite side to the hypotenuse for triangle \( ABC \) is \( \frac{4}{5} \).
• Point \( A' \) in triangle \( A'B'C' \ is actually opposite the side ( B'C' = 24\) (not 32).

Thus, none of the options are entirely correct, but B accurately states the ratio, although the specific association needs to be clarified.

Thank you for your patience, and I apologize for any confusion earlier.