Drag and drop the given set of measurements into the correct box to describe whether the measurements can describe the lengths of the three sides of a right triangle or not. Show your work.(3 points) 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. Could Be the Sides of a Right Triangle Cannot Be the Sides of a Right Triangle

Question

63 in, 16 in, 65in

4 in, 5in, 6in

29in, 20in, 21in

Answer 1

To determine if a set of measurements can describe the lengths of the three sides of a right triangle, we can use the Pythagorean theorem, which states that in a right triangle with legs \( a \) and \( b \) and hypotenuse \( c \), the following must hold true:

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

We need to identify which measurement in each set serves as the hypotenuse (the longest measurement), and then test if the equation holds true.

1. Measurements: 63 in, 16 in, 65 in

   • Hypotenuse: 65 in
   • Check: \( 16^2 + 63^2 = 256 + 3969 = 4225 \)
   • \( 65^2 = 4225 \)
   • Result: Could be the sides of a right triangle.

2. Measurements: 4 in, 5 in, 6 in

   • Hypotenuse: 6 in
   • Check: \( 4^2 + 5^2 = 16 + 25 = 41 \)
   • \( 6^2 = 36 \)
   • Result: Cannot be the sides of a right triangle.

3. Measurements: 29 in, 20 in, 21 in

   • Hypotenuse: 29 in
   • Check: \( 20^2 + 21^2 = 400 + 441 = 841 \)
   • \( 29^2 = 841 \)
   • Result: Could be the sides of a right triangle.

Now we can categorize the measurements:

• Could Be the Sides of a Right Triangle:

  • 63 in, 16 in, 65 in
  • 29 in, 20 in, 21 in

• Cannot Be the Sides of a Right Triangle:

  • 4 in, 5 in, 6 in

Therefore, the responses should be sorted as follows:

• Could Be the Sides of a Right Triangle:

  • 63 in, 16 in, 65 in
  • 29 in, 20 in, 21 in

• Cannot Be the Sides of a Right Triangle:

  • 4 in, 5 in, 6 in