Question

determine whether the triangle with the given side lengths is a right triangle.

1. 8, 15, 17
2. 11, 13, 17
3. 20, 21, 28
4. 5, 64, 65
5. 9, 12, 15

the lengths of two sides of a third side.
(and some right triangle diagrams with labeled sides: e.g., a right triangle with legs 26 and a, hypotenuse 38; another with leg 12, hypotenuse 18, leg b; etc.)

Explanation:

Step1: Solve for $a$ (Q4)

Use Pythagorean theorem: $a = \sqrt{38^2 - 25^2}$
$\sqrt{1444 - 625} = \sqrt{819} \approx 28.62$

Step2: Solve for $b$ (Q5)

Use Pythagorean theorem: $b = \sqrt{18^2 - 12^2}$
$\sqrt{324 - 144} = \sqrt{180} = 6\sqrt{5} \approx 13.42$

Step3: Solve for $c$ (Q3)

Use Pythagorean theorem: $c = \sqrt{7^2 + 24^2}$
$\sqrt{49 + 576} = \sqrt{625} = 25$

Step4: Solve for $c$ (Q6)

Use Pythagorean theorem: $c = \sqrt{2^2 + 8^2}$
$\sqrt{4 + 64} = \sqrt{68} = 2\sqrt{17} \approx 8.25$

Step5: Check Q7 (8,15,17)

Verify $8^2 + 15^2 = 17^2$
$64 + 225 = 289$, $289=289$: Right triangle

Step6: Check Q8 (20,21,28)

Verify $20^2 + 21^2 = 28^2$
$400 + 441 = 841$, $784
eq 841$: Not right triangle

Step7: Check Q9 (9,12,15)

Verify $9^2 + 12^2 = 15^2$
$81 + 144 = 225$, $225=225$: Right triangle

Step8: Check Q10 (11,13,17)

Verify $11^2 + 13^2 = 17^2$
$121 + 169 = 290$, $289
eq 290$: Not right triangle

Step9: Check Q11 (5,64,65)

Verify $5^2 + 64^2 = 65^2$
$25 + 4096 = 4121$, $4225
eq 4121$: Not right triangle

Answer:

1. $a = \sqrt{819} \approx 28.62$
2. $b = 6\sqrt{5} \approx 13.42$
3. $c = 25$
4. $c = 2\sqrt{17} \approx 8.25$
5. Right triangle
6. Not a right triangle
7. Right triangle
8. Not a right triangle
9. Not a right triangle