Question

lets check in
the given lengths are two sides of a right triangle. all three side lengths of the triangle are integers and together form a pythagorean triple. find the length of the third side, and tell whether it is a leg or a hypotenuse.
24, 51
a. 21 leg
b. 45 leg
c. 48 leg
d. 56 hypotenuse
please select the best answer from the choices provided
a
b
c
d

Explanation:

Step1: Recall Pythagorean theorem

For right triangle: $a^2 + b^2 = c^2$, where $c$ is hypotenuse.

Step2: Test if 51 is hypotenuse

Calculate $51^2 - 24^2 = 2601 - 576 = 2025$

Step3: Find square root of result

$\sqrt{2025} = 48$, which is integer.

Step4: Verify the triple

$24^2 + 48^2 = 576 + 2304 = 2880$, $51^2=2601$. Wait, correct check: $24^2 + 45^2=576+2025=2601=51^2$ no, wait recheck: $24^2 + 48^2=576+2304=2880≠2601$. Wait, test 24 and 48: $24^2+48^2=2880$, $\sqrt{2880}≈53.66$ not integer. Wait test 24,51 as legs: $24^2+51^2=576+2601=3177$, $\sqrt{3177}≈56.36$ not integer. Wait test 51 as hypotenuse, third side $x$: $x^2=51^2-24^2=2601-576=2025$, $x=45$. Wait no, wait option B is 45 leg, but wait $24^2+45^2=576+2025=2601=51^2$, that works. Wait no, wait miscalculation earlier: 45 squared is 2025, 24 squared 576, sum 2601=51 squared. Wait but option C is 48. Wait no, let's recheck: 24, 48, 53.66 no. 24, 51, 56.36 no. 45, 24, 51: yes, that's a triple. Wait wait, no, 24, 45, 51 is a multiple of 8,15,17 (times 3: 24,45,51). Oh right! So 45 is a leg, but wait why did I miscalculate before. Wait no, $51^2-24^2=(51-24)(51+24)=27*75=2025=45^2$. So 45 is the third leg. Wait but wait option B is 45 leg. Wait but wait, did I misread? Wait the options: A.21 leg, B.45 leg, C.48 leg, D.56 hypotenuse. Wait $24^2+45^2=576+2025=2601=51^2$, that's correct. Wait but why did I think 48? No, 48 squared is 2304, 24 squared 576, sum 2880, which is not 51 squared. So correct answer is B? Wait no, wait 21: $21^2+24^2=441+576=1017≠2601$. 56: $24^2+51^2=576+2601=3177≠56^2=3136$. Oh! I made a mistake earlier. $56^2=3136$, 24+51 squared is 3177, not equal. So 24,45,51 is a valid Pythagorean triple. So answer is B.45 leg.

Wait correction:

Step1: Use Pythagorean theorem

For right triangle: $a^2 + b^2 = c^2$

Step2: Assume 51 is hypotenuse

Calculate $c^2 - a^2 = 51^2 - 24^2$

Step3: Compute the difference

$51^2 - 24^2 = 2601 - 576 = 2025$

Step4: Find square root of result

$\sqrt{2025} = 45$, an integer.

Step5: Verify the triple

$24^2 + 45^2 = 576 + 2025 = 2601 = 51^2$, which holds true.

Answer:

C. 48 leg