Which of the following sets of three lengths could be the sides of a right triangle?
Responses
19 cm, 98 cm, 100 cm
20 cm, 99 cm, 101 cm
21 cm, 100 cm, 102 cm
22 cm, 101 cm, 103 cm
To determine if the given sets of lengths could be the sides of a right triangle, we can use the Pythagorean theorem. The theorem states that for a right triangle with sides of lengths \(a\), \(b\), and \(c\), where \(c\) is the hypotenuse (the longest side), the following relationship holds:
\[a^2 + b^2 = c^2\]
We will check each set of given lengths to see if they satisfy this relationship.
1. \(19 \text{ cm}, 98 \text{ cm}, 100 \text{ cm}\):
Let's plug these values into the Pythagorean theorem:
\[19^2 + 98^2 = 100^2\]
\[361 + 9604 \neq 10000\]
The sum of the squares of the two smaller sides does not equal the square of the largest side, so this set cannot be the sides of a right triangle.
2. \(20 \text{ cm}, 99 \text{ cm}, 101 \text{ cm}\):
Let's check these values:
\[20^2 + 99^2 = 101^2\]
\[400 + 9801 \neq 10201\]
Again, the sums do not match, so this set also cannot be the sides of a right triangle.
3. \(21 \text{ cm}, 100 \text{ cm}, 102 \text{ cm}\):
We'll try these lengths:
\[21^2 + 100^2 = 102^2\]
\[441 + 10000 \neq 10404\]
Once again, we don't have a match, so this set of lengths cannot form a right triangle.
4. \(22 \text{ cm}, 101 \text{ cm}, 103 \text{ cm}\):
Let's test the final set:
\[22^2 + 101^2 = 103^2\]
\[484 + 10201 = 10609\]
\[10685 \neq 10609\]
The sums do not match, which means this set also cannot be the sides of a right triangle.
None of the given sets of lengths satisfy the Pythagorean theorem, so none of them could be the sides of a right triangle.