Question

solve for x. leave your answer in simplest radical form.

Explanation:

Step1: Find the middle side

In the right triangle with legs 5 and the middle side (let's call it \( y \)), hypotenuse 8. Use Pythagorean theorem: \( y^2 + 5^2 = 8^2 \)
\( y^2 = 64 - 25 = 39 \)? Wait, no, wait. Wait, the lower triangle: base 3, height \( x \), hypotenuse \( y \). The upper triangle: legs 5 and \( y \), hypotenuse 8? Wait, no, the upper triangle has a right angle, so legs 5 and \( y \), hypotenuse 8? Wait, no, the figure: there are two right triangles. The lower one: right angle at bottom, legs 3 and \( x \), hypotenuse \( y \). The upper one: right angle at the left, legs 5 and \( y \), hypotenuse 8? Wait, no, the upper triangle's hypotenuse is 8? Wait, the upper triangle: one leg is 5, the other leg is \( y \), and hypotenuse is 8? Wait, no, maybe I got the triangles wrong. Wait, the figure: the left side is \( x \), bottom is 3, right angle at bottom. Then a line from bottom right to the top left, making a right angle at the top left (between 5 and \( y \)). So the upper triangle: right-angled at the left, legs 5 and \( y \), hypotenuse 8? Wait, no, the upper triangle's hypotenuse is 8? Wait, the side labeled 8 is the hypotenuse of the upper triangle? Wait, no, the side labeled 8 is a leg? Wait, no, let's re-examine.

Wait, the figure: two right triangles. The lower triangle: right angle at (0,0), base 3 (along x-axis), height \( x \) (along y-axis), hypotenuse \( y \) (from (0,0) to (3,x)? No, wait, the right angle is at the bottom, so (0,0) is bottom left, (3,0) is bottom right, (0,x) is top left, and (3,x) is connected to (0,x) with a line? No, there's a line from (3,0) to (0,x), which is the hypotenuse \( y \) of the lower triangle. Then from (0,x) to (a,b) with a right angle, and a line from (a,b) to (3,0) with length 8, and from (0,x) to (a,b) with length 5. Wait, maybe the upper triangle is right-angled at (0,x), so legs 5 and \( y \), hypotenuse 8. So \( 5^2 + y^2 = 8^2 \)? Wait, no, \( 5^2 + y^2 = 8^2 \) would be \( y^2 = 64 -25 = 39 \), but then the lower triangle: \( 3^2 + x^2 = y^2 \), so \( x^2 = y^2 -9 = 39 -9 = 30 \), so \( x = \sqrt{30} \)? Wait, no, that can't be. Wait, maybe I mixed up the triangles. Wait, maybe the upper triangle has legs 5 and 8? No, the side labeled 5 is a leg, 8 is a leg, and \( y \) is the hypotenuse? Wait, no, the right angle is at the left, so legs are 5 and \( y \), hypotenuse 8? No, that would be \( 5^2 + y^2 = 8^2 \), \( y^2 = 39 \), then lower triangle: \( 3^2 + x^2 = y^2 \), so \( x^2 = 39 -9 = 30 \), so \( x = \sqrt{30} \). Wait, but let's check again.

Wait, the correct approach: The two right triangles are similar? No, not necessarily. Wait, the key is that the hypotenuse of the lower right triangle (with legs 3 and \( x \)) is equal to the other leg of the upper right triangle (with legs 5 and that hypotenuse, and hypotenuse 8). So let's denote:

Let \( y \) be the length of the segment connecting the bottom right to the top left (the hypotenuse of the lower triangle). Then, in the upper right triangle (right-angled at the left), we have:

\( 5^2 + y^2 = 8^2 \)? Wait, no, that would mean \( y^2 = 64 -25 = 39 \). Then, in the lower right triangle (right-angled at the bottom), we have:

\( 3^2 + x^2 = y^2 \)

So \( x^2 = y^2 - 9 = 39 - 9 = 30 \)

Thus, \( x = \sqrt{30} \). Wait, but let's confirm the triangles.

Wait, the upper triangle: right angle at the left, so one leg is 5 (horizontal), one leg is \( y \) (vertical), and hypotenuse 8 (the side going to the top right). Then the lower triangle: right angle at the bottom, one leg is 3 (horizontal), one leg is \( x \) (vertical), and hypotenuse \( y \) (the side connecting bottom right to top left). So yes, that makes sense. So first, find \( y \) using the upper triangle: \( 5^2 + y^2 = 8^2 \)? Wait, no, wait: the upper triangle's legs are 5 and \( y \), hypotenuse 8? Wait, no, the side labeled 8 is a leg? Wait, no, the side labeled 8 is the hypotenuse of the upper triangle? Wait, no, the upper triangle has a right angle, so the two legs are 5 and \( y \), and the hypotenuse is 8. So Pythagorean theorem: \( 5^2 + y^2 = 8^2 \). Then \( y^2 = 64 -25 = 39 \). Then the lower triangle: legs 3 and \( x \), hypotenuse \( y \), so \( 3^2 + x^2 = y^2 \). So \( x^2 = 39 -9 = 30 \), so \( x = \sqrt{30} \). Wait, but let me check again. Wait, maybe the upper triangle's hypotenuse is 8, and one leg is 5, so the other leg is \( \sqrt{8^2 -5^2} = \sqrt{64 -25} = \sqrt{39} \). Then the lower triangle: legs 3 and \( x \), hypotenuse \( \sqrt{39} \), so \( x = \sqrt{(\sqrt{39})^2 - 3^2} = \sqrt{39 -9} = \sqrt{30} \). Yes, that's correct.

Step2: Solve for \( x \)

So \( x = \sqrt{30} \)

Answer:

\( \sqrt{30} \)