Question

find the exact value of x.
x =
do the side lengths form a pythagorean triple?
yes
no
(image of a right triangle with legs 9 and x, hypotenuse 24)

Explanation:

Step1: Apply Pythagorean theorem

For a right triangle, \( a^2 + b^2 = c^2 \), where \( c \) is hypotenuse, \( a,b \) are legs. Here, hypotenuse is \( 24 \), one leg is \( 9 \), the other leg is \( x \). So \( x^2 + 9^2 = 24^2 \).

Step2: Solve for \( x^2 \)

Calculate \( 24^2 = 576 \), \( 9^2 = 81 \). Then \( x^2 = 576 - 81 = 495 \).

Step3: Simplify \( x \)

\( x = \sqrt{495} = \sqrt{9\times55} = 3\sqrt{55} \). Wait, no, wait, maybe I mixed hypotenuse. Wait, the triangle: right angle, so legs are \( x \) and \( 9 \), hypotenuse is \( 24 \)? Wait, no, maybe the hypotenuse is \( x \)? Wait, the diagram: right triangle, one leg 9, hypotenuse 24? No, maybe the other leg is 24? Wait, no, the labels: the triangle has right angle, one leg 9, another leg 24? Wait, no, the text: "Find the exact value of x." The triangle: right angle, one leg 9, hypotenuse x? Wait, no, the original problem: the triangle has sides x, 9, and 24, with right angle. So let's recheck.

Wait, maybe I misread the hypotenuse. Let's assume the right triangle has legs 9 and x, hypotenuse 24? No, that can't be, because 9 and x are legs, hypotenuse 24. Then \( x^2 + 9^2 = 24^2 \). So \( x^2 = 576 - 81 = 495 \), so \( x = \sqrt{495} = 3\sqrt{55} \approx 22.25 \). But wait, maybe the hypotenuse is x, and one leg is 24, another is 9? Then \( 9^2 + 24^2 = x^2 \). Let's calculate that: \( 81 + 576 = 657 \), so \( x = \sqrt{657} = \sqrt{9\times73} = 3\sqrt{73} \). Wait, the diagram: the right angle is at the bottom, so the vertical leg is x, horizontal leg is 9, hypotenuse is 24? Or horizontal leg 9, vertical leg 24, hypotenuse x? Wait, the label "24" is on the hypotenuse? Or on a leg? The user's diagram: "24" is on the hypotenuse? Wait, the user's image: the triangle has right angle, one leg 9 (horizontal), hypotenuse 24 (the slant side), and vertical leg x. So then:

\( x^2 + 9^2 = 24^2 \)

\( x^2 = 24^2 - 9^2 = 576 - 81 = 495 \)

\( x = \sqrt{495} = \sqrt{9 \times 55} = 3\sqrt{55} \). But wait, 495 is 9*55, so that's correct. But wait, maybe the hypotenuse is x, and the other leg is 24. Let's check the problem again. The problem says "Find the exact value of x." The triangle: right angle, one leg 9, one leg 24, hypotenuse x. Then:

\( 9^2 + 24^2 = x^2 \)

\( 81 + 576 = x^2 \)

\( 657 = x^2 \)

\( x = \sqrt{657} = \sqrt{9 \times 73} = 3\sqrt{73} \). Wait, now I'm confused. Let's look at the diagram again. The user's image: the right triangle has right angle at the bottom, horizontal leg 9, vertical leg x, hypotenuse 24 (labeled 24). So vertical leg x, horizontal 9, hypotenuse 24. So Pythagorean theorem: \( x^2 + 9^2 = 24^2 \). So \( x^2 = 576 - 81 = 495 \), so \( x = \sqrt{495} = 3\sqrt{55} \). But 495 is 9*55, so that's correct. Alternatively, maybe the 24 is a leg. Let's see: if 24 is a leg, 9 is a leg, then hypotenuse x: \( x = \sqrt{9^2 + 24^2} = \sqrt{81 + 576} = \sqrt{657} = 3\sqrt{73} \). Now, which is it? The diagram: the hypotenuse is the slant side, so if 24 is the slant side, then x is vertical leg. But maybe the label "24" is on the vertical leg? Wait, the user's image: the triangle has right angle, horizontal leg 9 (labeled 9), hypotenuse 24 (labeled 24), vertical leg x (labeled x). So yes, hypotenuse 24, leg 9, leg x. So \( x^2 + 9^2 = 24^2 \). So \( x = \sqrt{24^2 - 9^2} = \sqrt{576 - 81} = \sqrt{495} = 3\sqrt{55} \). Wait, but 495 is 9*55, so that's correct. Alternatively, maybe I made a mistake in hypotenuse. Let's check the Pythagorean triple. 9, x, 24: are they a Pythagorean triple? A Pythagorean triple has integer sides. 9, 12, 15 is a triple (3-4-5 scaled by 3). 9, 24, x: 9=3*3, 24=3*8, so the triple would be 3,8,x/3. 3^2 +8^2=9+64=73, so x/3=√73, so x=3√73. Ah! So if the legs are 9 and 24, hypotenuse x, then x=3√73, and 9,24,3√73: but 3√73 is not integer, so not a Pythagorean triple. Wait, but if hypotenuse is 24, leg 9, then x=√(24²-9²)=√(576-81)=√495=3√55, also not integer. So maybe the diagram has hypotenuse x, legs 9 and 24. Then x=√(9²+24²)=√(81+576)=√657=3√73. Then, do 9,24,3√73 form a Pythagorean triple? No, because 3√73 is irrational. Wait, but maybe the problem has a typo, or I misread the hypotenuse. Wait, let's recalculate:

If hypotenuse is x, legs 9 and 24:

\( x^2 = 9^2 + 24^2 = 81 + 576 = 657 \)

\( x = \sqrt{657} = \sqrt{9 \times 73} = 3\sqrt{73} \)

If hypotenuse is 24, leg 9, leg x:

\( x^2 = 24^2 - 9^2 = 576 - 81 = 495 \)

\( x = \sqrt{495} = \sqrt{9 \times 55} = 3\sqrt{55} \)

Now, let's check the "Do the side lengths form a Pythagorean triple?" A Pythagorean triple consists of three positive integers a, b, c, such that a² + b² = c². So if x is 3√55 or 3√73, it's not integer, so the answer to "Do they form a Pythagorean triple?" is No. But let's confirm the first part: finding x. Let's assume the hypotenuse is x, legs 9 and 24 (maybe the diagram's 24 is a leg). Let's proceed with that, as 9 and 24 are legs, hypotenuse x.

Step1: Apply Pythagorean theorem (correcting hypotenuse as x)

For right triangle with legs \( a = 9 \), \( b = 24 \), hypotenuse \( c = x \), so \( x^2 = 9^2 + 24^2 \).

Step2: Calculate squares

\( 9^2 = 81 \), \( 24^2 = 576 \).

Step3: Sum and take square root

\( x^2 = 81 + 576 = 657 \), so \( x = \sqrt{657} = \sqrt{9 \times 73} = 3\sqrt{73} \). Wait, but maybe the hypotenuse is 24. Let's check both. Wait, the problem says "Find the exact value of x." Let's see the diagram again: the right angle is at the bottom, horizontal leg 9, vertical leg x, hypotenuse 24 (the slant side). So hypotenuse is 24, so \( x^2 + 9^2 = 24^2 \). Then \( x^2 = 576 - 81 = 495 \), so \( x = \sqrt{495} = 3\sqrt{55} \). Both are possible, but let's check the Pythagorean triple. A Pythagorean triple has integer sides. So if x is 3√55 (≈22.25) or 3√73 (≈25.63), neither is integer, so the answer to "Do they form a Pythagorean triple?" is No. But let's confirm the x value. Let's re-express the Pythagorean theorem correctly.

Correct approach: In a right triangle, the sum of the squares of the legs equals the square of the hypotenuse. So identify which side is the hypotenuse. The hypotenuse is the side opposite the right angle, so it's the longest side. So if 24 is longer than 9, then hypotenuse is 24, leg is 9, leg is x. Then x = √(24² - 9²) = √(576 - 81) = √495 = 3√55. If x is longer than 24, then hypotenuse is x, legs 9 and 24, so x = √(9² + 24²) = √657 = 3√73. Since 24 > 9, 24 is longer than 9, so hypotenuse is 24, so x is a leg. So x = 3√55. But 3√55 is the exact value.

Wait, but maybe the problem has a typo, and the hypotenuse is 25 instead of 24? Then 9,12,15 (no), 7,24,25: 7²+24²=49+576=625=25². But here it's 9, x,24. So 9² + x² =24² → x²=576-81=495→x=√495=3√55. So that's the exact value.

Answer:

For finding \( x \): \( 3\sqrt{55} \) (or if hypotenuse is \( x \), \( 3\sqrt{73} \); but based on hypotenuse being the longest side, \( x = 3\sqrt{55} \)). For the Pythagorean triple: No, because \( x \) is not an integer.

Wait, but let's redo the calculation with hypotenuse as x (maybe the diagram's 24 is a leg). Let's assume legs are 9 and 24, hypotenuse x. Then:

\( x^2 = 9^2 + 24^2 = 81 + 576 = 657 \)

\( x = \sqrt{657} = \sqrt{9 \times 73} = 3\sqrt{73} \)

Now, check if 9,24,3√73 are a Pythagorean triple: no, because 3√73 is not integer. If hypotenuse is 24, legs 9 and x: 9, x,24. x=3√55, not integer, so not a Pythagorean triple. So the answer to "Do the side lengths form a Pythagorean triple?" is No.

So first part: x = 3√55 (or 3√73, depending on hypotenuse). Let's confirm with the diagram: the right angle is at the bottom, so vertical leg x, horizontal leg 9, hypotenuse 24 (the slant side). So hypotenuse is 24, so x is vertical leg. So x = √(24² - 9²) = √495 = 3√55.

Final answer for x: \( 3\sqrt{55} \) (exact value), and the side lengths do not form a Pythagorean triple (answer No).