In each case, determine if the sides form a right triangle. If they do, place in the Yes column. If they do not, place in the No column.

Question

Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
Yes No

Responces:
9, 12, 14
6, 9, 12
5, 12, 13
3, 4, 5

Answer 1

To determine if the sides form a right triangle, we can use the Pythagorean theorem, which states that for a right triangle with sides \(a\), \(b\), and hypotenuse \(c\) (where \(c\) is the longest side), the following equation must hold true:

\[ a^2 + b^2 = c^2 \]

Let's evaluate each set of triangle sides:

1. 9, 12, 14:

   • Longest side is 14.
   • \(9^2 + 12^2 = 81 + 144 = 225\)
   • \(14^2 = 196\)
   • Does not satisfy \(a^2 + b^2 = c^2\)
   • No

2. 6, 9, 12:

   • Longest side is 12.
   • \(6^2 + 9^2 = 36 + 81 = 117\)
   • \(12^2 = 144\)
   • Does not satisfy \(a^2 + b^2 = c^2\)
   • No

3. 5, 12, 13:

   • Longest side is 13.
   • \(5^2 + 12^2 = 25 + 144 = 169\)
   • \(13^2 = 169\)
   • Satisfies \(a^2 + b^2 = c^2\)
   • Yes

4. 3, 4, 5:

   • Longest side is 5.
   • \(3^2 + 4^2 = 9 + 16 = 25\)
   • \(5^2 = 25\)
   • Satisfies \(a^2 + b^2 = c^2\)
   • Yes

Now, let's place the responses in the correct columns:

Yes:

• 5, 12, 13
• 3, 4, 5

No:

• 9, 12, 14
• 6, 9, 12