Question

in the given right triangle, find the side labeled x.
x =
diagram of a right triangle with one leg labeled ( x ), another leg labeled ( 85 ), and hypotenuse labeled ( 4x )

Explanation:

Step1: Apply Pythagorean theorem

In a right triangle, \(a^2 + b^2 = c^2\). Here, \(a = x\), \(b = x\) (wait, no, one leg is \(x\), hypotenuse is \(4x\)? Wait, no, the hypotenuse is the longest side. Wait, the sides: one leg is \(x\), another leg? Wait, no, the triangle has sides \(x\), 85, and \(4x\)? Wait, no, the right angle is between \(x\) and the other leg? Wait, no, the hypotenuse should be the longest. Wait, maybe I misread. Wait, the right triangle: legs are \(x\) and 85? No, the hypotenuse is \(4x\)? Wait, no, 85 is a side, \(4x\) is a side, \(x\) is a side. Wait, right triangle, so Pythagorean theorem: \(x^2 + 85^2=(4x)^2\)? Wait, no, maybe the legs are \(x\) and 85, hypotenuse \(4x\)? Wait, no, 4x must be hypotenuse because it's longer? Wait, no, 85 is a side, \(x\) is a leg, \(4x\) is hypotenuse? Wait, let's check.

Wait, the right triangle: right angle, so two legs and hypotenuse. Let's assume the legs are \(x\) and 85, hypotenuse \(4x\)? No, that can't be, because \(4x\) would be hypotenuse, so \(x^2 + 85^2=(4x)^2\). Wait, no, maybe the legs are \(x\) and another leg, hypotenuse 85? Wait, the diagram: one side is 85, one leg is \(x\), hypotenuse \(4x\)? Wait, the problem says "find the side labeled \(x\)". Let's re-express.

Wait, the triangle: right-angled, so Pythagorean theorem: \(x^2 + (other leg)^2 = hypotenuse^2\). From the diagram, the sides are \(x\), 85, and \(4x\). Wait, maybe the legs are \(x\) and 85, hypotenuse \(4x\)? No, that would mean \(x^2 + 85^2 = (4x)^2\). Let's solve that.

\(x^2 + 7225 = 16x^2\)

Subtract \(x^2\) from both sides:

\(7225 = 15x^2\)

Then \(x^2 = 7225 / 15\)? Wait, that's not an integer. Wait, maybe I got the sides wrong. Wait, maybe the hypotenuse is 85, and one leg is \(x\), the other leg is \(4x\)? Wait, that makes more sense, because hypotenuse is the longest side, so 85 should be hypotenuse if \(4x\) is a leg. Wait, let's check: if hypotenuse is 85, legs are \(x\) and \(4x\), then Pythagorean theorem: \(x^2 + (4x)^2 = 85^2\)

Yes, that makes sense, because 4x would be a leg, and 85 hypotenuse. Let's do that.

Step1: Apply Pythagorean theorem

For right triangle, \(a^2 + b^2 = c^2\), where \(c\) is hypotenuse. Here, \(a = x\), \(b = 4x\), \(c = 85\). So:

\(x^2 + (4x)^2 = 85^2\)

Step2: Simplify the equation

Expand \((4x)^2 = 16x^2\), so:

\(x^2 + 16x^2 = 7225\)

Combine like terms:

\(17x^2 = 7225\)

Step3: Solve for \(x^2\)

Divide both sides by 17:

\(x^2 = \frac{7225}{17}\)

Calculate \(7225 ÷ 17 = 425\)? Wait, 17*425 = 7225? Let's check: 17*400=6800, 17*25=425, so 6800+425=7225. Yes. So \(x^2 = 425\)? Wait, no, wait, 17*425=7225? Wait, 425*17: 400*17=6800, 25*17=425, 6800+425=7225. Yes. So \(x^2 = 425\)? Wait, no, wait, 85^2 is 7225, correct. Then 17x²=7225, so x²=7225/17=425? Wait, no, 7225 divided by 17: 17*425=7225? Wait, 425*17: 425*10=4250, 425*7=2975, 4250+2975=7225. Yes. So x²=425? Wait, but 425 is 25*17, so x=√(25*17)=5√17≈20.6155. But that seems odd. Wait, maybe I mixed up the hypotenuse. Wait, maybe the hypotenuse is 4x, and one leg is 85, the other is x. Then:

\(x^2 + 85^2 = (4x)^2\)

\(x^2 + 7225 = 16x^2\)

\(7225 = 15x^2\)

\(x^2 = 7225 / 15 ≈ 481.666\), which is not a perfect square. So that's unlikely. So probably the hypotenuse is 85, legs x and 4x. Then x=5√17? Wait, but maybe I made a mistake. Wait, let's check again.

Wait, the diagram: the right angle is at the top, so the two legs are the vertical (x) and horizontal (let's say y), and the hypotenuse is 85? No, the other side is 4x. Wait, maybe the sides are x (leg), 4x (leg), and 85 (hypotenuse). Then Pythagorean theorem: x² + (4x)² = 85². So x² + 16x² = 7225 → 17x²=7225 → x²=7225/17=425 → x=√425=5√17≈20.6155. But maybe the problem has integer solution. Wait, maybe I misread the side: is the side 85 or 84? Wait, the user wrote 85. Alternatively, maybe the hypotenuse is 4x, and one leg is 85, the other is x, but then 4x must be longer than 85, so x>85/4=21.25. Then x² + 85² = (4x)² → 15x²=7225 → x²=7225/15≈481.666, x≈21.94. Not integer. Alternatively, maybe the legs are 85 and x, hypotenuse 4x, but 4x>85 and 4x>x, so 4x>85. Then x² + 85² = (4x)² → 15x²=7225 → x²=7225/15=481.666, x≈21.94. Not integer. Alternatively, maybe the side is 84 instead of 85. Let's check: 84²=7056. Then 17x²=7056 → x²=7056/17≈415.05, not integer. Alternatively, 85 is correct, and the answer is 5√17, but maybe I made a mistake. Wait, maybe the triangle is isoceles? No, the legs are x and 4x, so not isoceles. Wait, maybe the problem is written as 4x is a leg, 85 is hypotenuse, and the other leg is x. So Pythagorean theorem: x² + (4x)² = 85² → 17x²=7225 → x²=425 → x=√425=5√17≈20.6155. So that's the solution.

Answer:

\(5\sqrt{17}\) (or approximately 20.62)