Question

A Pythagorean triple consists of three positive integers that work with the Pythagorean Theorem.
Find the missing value that be a Pythagorean Triple for a right triangle that has a length of 20 and 99.
Find the missing value that be a Pythagorean Triple for a right triangle that has a length of 180 and 181.

Answer 1

A Pythagorean triple consists of three positive integers \( a \), \( b \), and \( c \) such that \( a^2 + b^2 = c^2 \). In each case, we will identify the missing value (which we denote \( c \)).

### Case 1: Lengths 20 and 99

Let's denote the two known sides as \( a = 20 \) and \( b = 99 \). We want to find \( c \).

Using the Pythagorean theorem:

\[
c^2 = a^2 + b^2
\]
\[
c^2 = 20^2 + 99^2
\]
Calculating \( 20^2 \) and \( 99^2 \):

\[
20^2 = 400
\]
\[
99^2 = 9801
\]
Adding these together:

\[
c^2 = 400 + 9801 = 10201
\]
Now take the square root to find \( c \):

\[
c = \sqrt{10201} = 101
\]

### Case 2: Lengths 180 and 181

Let’s denote the two known sides as \( a = 180 \) and \( b = 181 \). We want to find \( c \).

Using the Pythagorean theorem again:

\[
c^2 = a^2 + b^2
\]
\[
c^2 = 180^2 + 181^2
\]
Calculating \( 180^2 \) and \( 181^2 \):

\[
180^2 = 32400
\]
\[
181^2 = 32761
\]
Adding these together:

\[
c^2 = 32400 + 32761 = 65161
\]
Now take the square root to find \( c \):

\[
c = \sqrt{65161} = 255
\]

### Summary of the Results:

1. For the lengths 20 and 99, the missing value \( c \) is **101**.
2. For the lengths 180 and 181, the missing value \( c \) is **255**.