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 includes 183 using the given Pythagorean triple (11, 60, 61), we can use the property that if \((a, b, c)\) is a Pythagorean triple, then multiplying each element by a constant \(k\) will yield a new Pythagorean triple.

First, let's calculate the factor \(k\) such that one of the numbers in the new triple equals 183. We can look for \(k\) such that:

1. \(k \cdot 11 = 183\)
2. \(k \cdot 60 = 183\)
3. \(k \cdot 61 = 183\)

Starting with the first equation:

\[
k = \frac{183}{11} \approx 16.636
\]

Next, for the second equation:

\[
k = \frac{183}{60} \approx 3.05
\]

Now for the third equation:

\[
k = \frac{183}{61} \approx 3
\]

The most suitable option is \(k = 3\), which keeps the values as whole numbers. Now, let's compute the new values of the Pythagorean triple:

1. \(a = k \cdot 11 = 3 \cdot 11 = 33\)
2. \(b = k \cdot 60 = 3 \cdot 60 = 180\)
3. \(c = k \cdot 61 = 3 \cdot 61 = 183\)

Thus, the new Pythagorean triple is:

\[
(33, 180, 183)
\]

Hence, the new Pythagorean triple that contains 183 is \((33, 180, 183)\).