Question

which measurements could not represent the side lengths of a right triangle? 84 cm, 288 cm, 300 cm; 595 cm, 618 cm, 834 cm; 80 cm, 84 cm, 116 cm; 33 cm, 44 cm, 55 cm

Explanation:

To determine which set of side lengths does not represent a right triangle, we use the Pythagorean theorem, \(a^2 + b^2 = c^2\), where \(c\) is the longest side.

Step 1: Check 84 cm, 288 cm, 300 cm

Calculate \(84^2 + 288^2\) and \(300^2\):
\(84^2 = 7056\), \(288^2 = 82944\), so \(84^2 + 288^2 = 7056 + 82944 = 90000\)
\(300^2 = 90000\)
Since \(84^2 + 288^2 = 300^2\), this is a right triangle.

Step 2: Check 595 cm, 618 cm, 834 cm

Calculate \(595^2 + 618^2\) and \(834^2\):
\(595^2 = 354025\), \(618^2 = 381924\), so \(595^2 + 618^2 = 354025 + 381924 = 735949\)
\(834^2 = 695556\)
Since \(595^2 + 618^2
eq 834^2\), this is not a right triangle.

Step 3: Check 80 cm, 84 cm, 116 cm

Calculate \(80^2 + 84^2\) and \(116^2\):
\(80^2 = 6400\), \(84^2 = 7056\), so \(80^2 + 84^2 = 6400 + 7056 = 13456\)
\(116^2 = 13456\)
Since \(80^2 + 84^2 = 116^2\), this is a right triangle.

Step 4: Check 33 cm, 44 cm, 55 cm

Calculate \(33^2 + 44^2\) and \(55^2\):
\(33^2 = 1089\), \(44^2 = 1936\), so \(33^2 + 44^2 = 1089 + 1936 = 3025\)
\(55^2 = 3025\)
Since \(33^2 + 44^2 = 55^2\), this is a right triangle.

Answer:

595 cm, 618 cm, 834 cm