Question

Given that
(11,60, 61) is a Pythagorean triple,
use
(11, 60, 61) to find a new Pythagorean triple that contains 183. (1 point)
The new Pythagorean triple is (
,183).

Answer 1

To find a new Pythagorean triple that contains 183 using the original triple (11, 60, 61), we can use a scaling factor.

First, notice that the original triple represents the relationship \(11^2 + 60^2 = 61^2\). To create a new Pythagorean triple, we can scale the values in the original triple by a factor \(k\).

To find a suitable scaling factor \(k\) such that one of the values becomes 183, we can set one of the components equal to 183:

1. Set \(11k = 183\):
\[
k = \frac{183}{11} = 16.6364 \quad (\text{not an integer, discard})
\]

2. Set \(60k = 183\):
\[
k = \frac{183}{60} = 3.05 \quad (\text{not an integer, discard})
\]

3. Set \(61k = 183\):
\[
k = \frac{183}{61} = 3 \quad (\text{integer, keep})
\]

Now we scale the entire triple (11, 60, 61) by the integer \(k = 3\):

\[
(11 \times 3, 60 \times 3, 61 \times 3) = (33, 180, 183)
\]

Therefore, the new Pythagorean triple that contains 183 is:

\[
(33, 183)
\]

We can check this as follows:

\[
33^2 + 180^2 = 1089 + 32400 = 33489
\]
\[
183^2 = 33489
\]

Since both sides are equal, the new Pythagorean triple is confirmed.

So, the final answer is:

The new Pythagorean triple is \( (33, 183) \).