Question

tell whether each triangle with the given side lengths is a right triangle.

1. 11 cm, 60 cm, 61 cm
2. 5 ft, 12 ft, 15 ft
3. 9 in., 15 in., 17 in.
4. 15 m, 36 m, 39 m
5. 20 mm, 30 mm, 40 mm
6. 20 cm, 48 cm, 52 cm
7. 18.5 ft, 6 ft, 17.5 ft
8. 2 mi, 1.5 mi, 2.5 mi
9. 35 in., 45 in., 55 in.
10. 25 cm, 14 cm, 23 cm

Answer:

To determine if a triangle with given side lengths is a right triangle, we use the Pythagorean theorem, which states that for a right triangle with side lengths \(a\), \(b\) (the legs), and \(c\) (the hypotenuse, the longest side), \(a^2 + b^2 = c^2\) must hold. We will check each problem:

Problem 5: \(11\) cm, \(60\) cm, \(61\) cm

Step 1: Identify \(a\), \(b\), \(c\)

Let \(a = 11\), \(b = 60\), \(c = 61\) (since \(61\) is the longest side).

Step 2: Check \(a^2 + b^2 = c^2\)

Calculate \(a^2 + b^2\):
\(11^2 + 60^2 = 121 + 3600 = 3721\)

Calculate \(c^2\):
\(61^2 = 3721\)

Since \(11^2 + 60^2 = 61^2\), the triangle is a right triangle.

Problem 6: \(5\) ft, \(12\) ft, \(15\) ft

Step 1: Identify \(a\), \(b\), \(c\)

Let \(a = 5\), \(b = 12\), \(c = 15\) (since \(15\) is the longest side).

Step 2: Check \(a^2 + b^2 = c^2\)

Calculate \(a^2 + b^2\):
\(5^2 + 12^2 = 25 + 144 = 169\)

Calculate \(c^2\):
\(15^2 = 225\)

Since \(169
eq 225\), the triangle is not a right triangle.

Problem 7: \(9\) in, \(15\) in, \(17\) in

Step 1: Identify \(a\), \(b\), \(c\)

Let \(a = 9\), \(b = 15\), \(c = 17\) (since \(17\) is the longest side).

Step 2: Check \(a^2 + b^2 = c^2\)

Calculate \(a^2 + b^2\):
\(9^2 + 15^2 = 81 + 225 = 306\)

Calculate \(c^2\):
\(17^2 = 289\)

Since \(306
eq 289\), the triangle is not a right triangle.

Problem 8: \(15\) m, \(36\) m, \(39\) m

Step 1: Identify \(a\), \(b\), \(c\)

Let \(a = 15\), \(b = 36\), \(c = 39\) (since \(39\) is the longest side).

Step 2: Check \(a^2 + b^2 = c^2\)

Calculate \(a^2 + b^2\):
\(15^2 + 36^2 = 225 + 1296 = 1521\)

Calculate \(c^2\):
\(39^2 = 1521\)

Since \(15^2 + 36^2 = 39^2\), the triangle is a right triangle.

Problem 9: \(20\) mm, \(30\) mm, \(40\) mm

Step 1: Identify \(a\), \(b\), \(c\)

Let \(a = 20\), \(b = 30\), \(c = 40\) (since \(40\) is the longest side).

Step 2: Check \(a^2 + b^2 = c^2\)

Calculate \(a^2 + b^2\):
\(20^2 + 30^2 = 400 + 900 = 1300\)

Calculate \(c^2\):
\(40^2 = 1600\)

Since \(1300
eq 1600\), the triangle is not a right triangle.

Problem 10: \(20\) cm, \(48\) cm, \(52\) cm

Step 1: Identify \(a\), \(b\), \(c\)

Let \(a = 20\), \(b = 48\), \(c = 52\) (since \(52\) is the longest side).

Step 2: Check \(a^2 + b^2 = c^2\)

Calculate \(a^2 + b^2\):
\(20^2 + 48^2 = 400 + 2304 = 2704\)

Calculate \(c^2\):
\(52^2 = 2704\)

Since \(20^2 + 48^2 = 52^2\), the triangle is a right triangle.

Problem 11: \(18.5\) ft, \(6\) ft, \(17.5\) ft

Step 1: Identify \(a\), \(b\), \(c\)

Let \(a = 6\), \(b = 17.5\), \(c = 18.5\) (since \(18.5\) is the longest side).

Step 2: Check \(a^2 + b^2 = c^2\)

Calculate \(a^2 + b^2\):
\(6^2 + 17.5^2 = 36 + 306.25 = 342.25\)

Calculate \(c^2\):
\(18.5^2 = 342.25\)

Since \(6^2 + 17.5^2 = 18.5^2\), the triangle is a right triangle.

Problem 12: \(2\) mi, \(1.5\) mi, \(2.5\) mi

Step 1: Identify \(a\), \(b\), \(c\)

Let \(a = 1.5\), \(b = 2\), \(c = 2.5\) (since \(2.5\) is the longest side).

Step 2: Check \(a^2 + b^2 = c^2\)

Calculate \(a^2 + b^2\):
\(1.5^2 + 2^2 = 2.25 + 4 = 6.25\)

Calculate \(c^2\):
\(2.5^2 = 6.25\)

Since \(1.5^2 + 2^2 = 2.5^2\), the triangle is a right triangle.

Problem 13: \(35\) in, \(45\) in, \(55\) in

Step 1: Identify \(a\), \(b\), \(c\)

Let \(a = 35\), \(b = 45\), \(c = 55\) (since \(55\) is the longest side).

Step 2: Check \(a^2 + b^2 = c^2\)

Calculate \(a^2 + b^2\):
\(35^2 + 45^2 = 1225 + 2025 = 3250\)

Calculate \(c^2\):
\(55^2 = 3025\)

Since \(3250
eq 3025\), the triangle is not a right triangle.

Problem 14: \(25\) cm, \(14\) cm, \(23\) cm

Step 1: Identify \(a\), \(b\), \(c\)

Let \(a = 14\), \(b = 23\), \(c = 25\) (since \(25\) is the longest side).

Step 2: Check \(a^2 + b^2 = c^2\)

Calculate \(a^2 + b^2\):
\(14^2 + 23^2 = 196 + 529 = 725\)

Calculate \(c^2\):
\(25^2 = 625\)

Since \(725
eq 625\), the triangle is not a right triangle.

Final Answers:

1. \(\boldsymbol{\text{Yes}}\)
2. \(\boldsymbol{\text{No}}\)
3. \(\boldsymbol{\text{No}}\)
4. \(\boldsymbol{\text{Yes}}\)
5. \(\boldsymbol{\text{No}}\)
6. \(\boldsymbol{\text{Yes}}\)
7. \(\boldsymbol{\text{Yes}}\)
8. \(\boldsymbol{\text{Yes}}\)
9. \(\boldsymbol{\text{No}}\)
10. \(\boldsymbol{\text{No}}\)