Let's analyze the triangles using the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (longest side) is equal to the sum of the squares of the lengths of the other two sides. That is, $c^2 = a^2 + b^2$, where $c$ is the hypotenuse.

1) Triangle ABC

• $AB = 4 \, \text{ft}$
• $BC = 3 \, \text{ft}$
• $AC = 5 \, \text{ft}$

Applying the Pythagorean theorem:

$c^2 = a^2 + b^2$ $5^2 = 4^2 + 3^2$ $25 = 16 + 9$ $25 = 25$

This triangle is a right triangle.

2) Triangle EFG

• $EF = 8 \, \text{yd}$
• $FG = 5 \, \text{yd}$
• $EG = 9 \, \text{yd}$

Applying the Pythagorean theorem:

$9^2 = 8^2 + 5^2$ $81 = 64 + 25$ $81 = 89$

This triangle is not a right triangle.

3) Triangle UVW

• $UV = 7 \, \text{in}$
• $VW = 9 \, \text{in}$
• $UW$ is the hypotenuse (needs calculation).

Using the Pythagorean theorem:

$c^2 = a^2 + b^2$ $c^2 = 7^2 + 9^2$ $c^2 = 49 + 81$ $c^2 = 130$ $c = \sqrt{130} \approx 11.4 \, \text{in}$

This triangle is not a right triangle because the hypotenuse is not an integer and does not match the given side lengths.

4) Triangle LMN

• $LM = 12 \, \text{ft}$
• $MN = 9 \, \text{ft}$
• $LN = 15 \, \text{ft}$

Applying the Pythagorean theorem:

$15^2 = 12^2 + 9^2$ $225 = 144 + 81$ $225 = 225$

This triangle is a right triangle.

5) Triangle XYZ

• $XY = 12 \, \text{ft}$
• $YZ = 16 \, \text{ft}$
• $XZ = 20 \, \text{ft}$

Applying the Pythagorean theorem:

$20^2 = 12^2 + 16^2$ $400 = 144 + 256$ $400 = 400$

This triangle is a right triangle.

6) Triangle PQR

• $PQ = 15 \, \text{in}$
• $QR = 20 \, \text{in}$
• $PR = 25 \, \text{in}$

Applying the Pythagorean theorem:

$25^2 = 15^2 + 20^2$ $625 = 225 + 400$ $625 = 625$

This triangle is a right triangle.

Let me know if you want further details on any of these, or explanations for how to approach each case.

5 Related Questions:

1. How is the Pythagorean theorem derived?
2. Can all triangles be tested using the Pythagorean theorem?
3. What is the difference between acute, obtuse, and right triangles?
4. How can we calculate angles in triangles using trigonometry?
5. How do you handle cases where side lengths are decimals or involve square roots?

Tip:

Always check which side is the hypotenuse (the longest side) when applying the Pythagorean theorem; it helps avoid errors in calculations.