Question

Look at the squares of the side lengths in the table. Take a moment to consider whether there is a relationship between the squares of the legs, a
2
and b
2
, and the square of the hypotenuse, c
2
. Are any of the following relationships true?
a = 6b = 3 c =
45
​

a
2
b
2
c
2

36 9 45

a
2
+b
2
=c
2


a
2
−b
2
=c
2


a
2
·b
2
=c
2


None of these relationships are true.

Answer 1

To determine if any of the given relationships are true, let's analyze the squares of the side lengths provided:

1. We have:
- \( a = 6 \)
- \( b = 3 \)
- \( c = 45 \)

2. Now we can calculate the squares of \( a \), \( b \), and \( c \):
- \( a^2 = 6^2 = 36 \)
- \( b^2 = 3^2 = 9 \)
- \( c^2 = 45^2 = 2025 \)

3. Let's evaluate each of the proposed relationships:

**a. \( a^2 + b^2 = c^2 \)**:
\[
a^2 + b^2 = 36 + 9 = 45
\]
This is **not true** since \( c^2 = 2025 \).

**b. \( a^2 - b^2 = c^2 \)**:
\[
a^2 - b^2 = 36 - 9 = 27
\]
This is **not true** since \( c^2 = 2025 \).

**c. \( a^2 \cdot b^2 = c^2 \)**:
\[
a^2 \cdot b^2 = 36 \cdot 9 = 324
\]
This is **not true** since \( c^2 = 2025 \).

Since none of the relationships \( a^2 + b^2 = c^2 \), \( a^2 - b^2 = c^2 \), or \( a^2 \cdot b^2 = c^2 \) are true based on the computations, the conclusion is:

**None of these relationships are true.**