Question

determine whether a triangle with the given side lengths is a right triangle.
side lengths \tright triangle \tnot a right triangle \tnot enough information
(a) 9, 40, 41 \t○ \t○ \t○
(b) 10, 24, 26 \t○ \t○ \t○
(c) 12, 16, 21 \t○ \t○ \t○
(d) 6, 7, 9 \t○ \t○ \t○

Explanation:

To determine if a triangle with given side lengths is a right triangle, we use the Pythagorean theorem, \(a^2 + b^2 = c^2\), where \(c\) is the longest side. Let's analyze each option:

Part (a): 9, 40, 41

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

Let \(a = 9\), \(b = 40\), \(c = 41\) (since 41 is the longest side).

Step 2: Check Pythagorean theorem

Calculate \(a^2 + b^2\) and \(c^2\):
\(a^2 + b^2 = 9^2 + 40^2 = 81 + 1600 = 1681\)
\(c^2 = 41^2 = 1681\)
Since \(a^2 + b^2 = c^2\), this is a right triangle.

Part (b): 10, 24, 26

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

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

Step 2: Check Pythagorean theorem

\(a^2 + b^2 = 10^2 + 24^2 = 100 + 576 = 676\)
\(c^2 = 26^2 = 676\)
Since \(a^2 + b^2 = c^2\), this is a right triangle.

Part (c): 12, 16, 21

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

Let \(a = 12\), \(b = 16\), \(c = 21\) (21 is the longest side).

Step 2: Check Pythagorean theorem

\(a^2 + b^2 = 12^2 + 16^2 = 144 + 256 = 400\)
\(c^2 = 21^2 = 441\)
Since \(400
eq 441\) (\(a^2 + b^2
eq c^2\)), this is not a right triangle.

Part (d): 6, 7, 9

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

Let \(a = 6\), \(b = 7\), \(c = 9\) (9 is the longest side).

Step 2: Check Pythagorean theorem

\(a^2 + b^2 = 6^2 + 7^2 = 36 + 49 = 85\)
\(c^2 = 9^2 = 81\)
Since \(85
eq 81\) (\(a^2 + b^2
eq c^2\)), this is not a right triangle.

Final Answers:

• (a) Right triangle
• (b) Right triangle
• (c) Not a right triangle
• (d) Not a right triangle

To mark the table:

• For (a): Select "Right triangle"
• For (b): Select "Right triangle"
• For (c): Select "Not a right triangle"
• For (d): Select "Not a right triangle"

Answer:

To determine if a triangle with given side lengths is a right triangle, we use the Pythagorean theorem, \(a^2 + b^2 = c^2\), where \(c\) is the longest side. Let's analyze each option:

Part (a): 9, 40, 41

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

Let \(a = 9\), \(b = 40\), \(c = 41\) (since 41 is the longest side).

Step 2: Check Pythagorean theorem

Calculate \(a^2 + b^2\) and \(c^2\):
\(a^2 + b^2 = 9^2 + 40^2 = 81 + 1600 = 1681\)
\(c^2 = 41^2 = 1681\)
Since \(a^2 + b^2 = c^2\), this is a right triangle.

Part (b): 10, 24, 26

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

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

Step 2: Check Pythagorean theorem

\(a^2 + b^2 = 10^2 + 24^2 = 100 + 576 = 676\)
\(c^2 = 26^2 = 676\)
Since \(a^2 + b^2 = c^2\), this is a right triangle.

Part (c): 12, 16, 21

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

Let \(a = 12\), \(b = 16\), \(c = 21\) (21 is the longest side).

Step 2: Check Pythagorean theorem

\(a^2 + b^2 = 12^2 + 16^2 = 144 + 256 = 400\)
\(c^2 = 21^2 = 441\)
Since \(400
eq 441\) (\(a^2 + b^2
eq c^2\)), this is not a right triangle.

Part (d): 6, 7, 9

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

Let \(a = 6\), \(b = 7\), \(c = 9\) (9 is the longest side).

Step 2: Check Pythagorean theorem

\(a^2 + b^2 = 6^2 + 7^2 = 36 + 49 = 85\)
\(c^2 = 9^2 = 81\)
Since \(85
eq 81\) (\(a^2 + b^2
eq c^2\)), this is not a right triangle.

Final Answers:

• (a) Right triangle
• (b) Right triangle
• (c) Not a right triangle
• (d) Not a right triangle

To mark the table:

• For (a): Select "Right triangle"
• For (b): Select "Right triangle"
• For (c): Select "Not a right triangle"
• For (d): Select "Not a right triangle"