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

which measurements could represent the side lengths in feet of a right triangle?
a 10 ft, 24 ft, 26 ft
b 14 ft, 14 ft, 14 ft
c 3 ft, 3 ft, 18 ft
d 2 ft, 3 ft, 5 ft

Explanation:

To determine which measurements represent the side lengths of a right triangle, we use the Pythagorean theorem, \(a^2 + b^2 = c^2\), where \(c\) is the longest side. We check each option:

Option A: 10 ft, 24 ft, 26 ft

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

Let \(a = 10\), \(b = 24\), \(c = 26\) (since 26 is the longest).

Step 2: Calculate \(a^2 + b^2\)

\(a^2 + b^2 = 10^2 + 24^2 = 100 + 576 = 676\)

Step 3: Calculate \(c^2\)

\(c^2 = 26^2 = 676\)
Since \(10^2 + 24^2 = 26^2\), this satisfies the Pythagorean theorem.

Option B: 14 ft, 14 ft, 14 ft

This is an equilateral triangle (all sides equal), not a right triangle (right triangles have one 90° angle, equilateral have 60° angles).

Option C: 3 ft, 3 ft, 18 ft

Check the triangle inequality: \(3 + 3 = 6\), which is less than 18. So, this is not a valid triangle (let alone a right triangle).

Option D: 2 ft, 3 ft, 5 ft

Check the triangle inequality: \(2 + 3 = 5\), which is not greater than 5. So, this is not a valid triangle.

Answer:

A. 10 ft, 24 ft, 26 ft