Question

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

Explanation:

Step1: Find the height (h) of the left triangle

Let the height of the left right - triangle (with hypotenuse 10 and base 4) be \( h \). Using the Pythagorean theorem \( a^{2}+b^{2}=c^{2} \), where \( c = 10 \) and \( a = 4 \), we have \( h^{2}+4^{2}=10^{2} \).
So \( h^{2}=10^{2}-4^{2}=100 - 16=84 \).

Step2: Use the height to find x in the right triangle

Now, consider the right - triangle on the right with hypotenuse 8 and height \( h \) (where \( h^{2}=84 \)) and base \( x \). Using the Pythagorean theorem \( x^{2}+h^{2}=8^{2} \).
Substitute \( h^{2}=84 \) into the equation: \( x^{2}+84 = 64 \)? Wait, no, that can't be. Wait, I made a mistake. Wait, the left triangle: the leg is 4, hypotenuse 10, so the other leg (height) \( h=\sqrt{10^{2}-4^{2}}=\sqrt{100 - 16}=\sqrt{84} \). Then the right triangle: hypotenuse 8, one leg is \( h \)? No, wait, the right triangle has legs \( x \) and \( h \), and hypotenuse 8? Wait, no, the right triangle has hypotenuse 8, one leg is \( x \), and the other leg is the same height \( h \) as the left triangle? Wait, no, the left triangle is a right triangle with legs 4 and \( h \), hypotenuse 10. The right triangle is a right triangle with legs \( x \) and \( h \), hypotenuse 8? Wait, that would mean \( h^{2}+x^{2}=8^{2} \), but we know \( h^{2}=10^{2}-4^{2}=84 \), so \( 84+x^{2}=64 \), which gives \( x^{2}=64 - 84=- 20 \), which is impossible. Wait, I misidentified the triangles. Wait, the left triangle: the base is 4, hypotenuse 10, right - angled at the bottom. The right triangle: right - angled at the right, with hypotenuse 8, and the vertical side is the same as the vertical side of the left triangle. Wait, no, the vertical segment is common to both triangles. Let's re - define: Let the vertical segment be \( y \). For the left triangle (right - angled at the bottom), \( y^{2}+4^{2}=10^{2}\), so \( y^{2}=100 - 16 = 84 \), so \( y=\sqrt{84} \). For the right triangle (right - angled at the right), \( y^{2}+x^{2}=8^{2}\)? No, that would be \( 84+x^{2}=64 \), which is wrong. Wait, no, the right triangle has hypotenuse 8, one leg is \( x \), and the other leg is \( y \), but that would mean \( x^{2}+y^{2}=8^{2} \), but \( y^{2}=84>64 \), which is impossible. So I must have misseen the triangle. Wait, maybe the right triangle has hypotenuse 8, one leg is \( x \), and the other leg is the vertical side, but the vertical side is a leg of the left triangle. Wait, no, maybe the left triangle: leg 4, hypotenuse 10, so vertical leg \( y=\sqrt{10^{2}-4^{2}}=\sqrt{84} \). Then the right triangle: leg \( x \), vertical leg \( y \), and hypotenuse 8? No, that can't be. Wait, maybe the right triangle has hypotenuse 8, one leg is \( x \), and the other leg is \( y \), but \( y \) is a leg, so \( x^{2}+y^{2}=8^{2} \), but \( y^{2}=84 \), so \( x^{2}=64 - 84=-20 \), which is impossible. So I must have mixed up the triangles. Wait, maybe the left triangle: hypotenuse 10, one leg 4, the other leg \( y \). The right triangle: hypotenuse 8, one leg \( y \), and the other leg \( x \). Wait, no, that would be \( y^{2}+x^{2}=8^{2} \), but \( y^{2}=10^{2}-4^{2}=84 \), so \( x^{2}=64 - 84=-20 \), which is wrong. Wait, maybe the right triangle has hypotenuse 8, one leg is \( x \), and the other leg is the same as the vertical leg of the left triangle, but the left triangle's vertical leg is longer than 8, which can't be. Wait, I think I made a mistake in the diagram. Wait, the left triangle: base 4, hypotenuse 10, right - angled at the bottom. The right triangle: right - angled at the right, with hypotenuse 8, and the horizontal leg is \( x \), and the vertical leg is the same as the vertical leg of the left triangle. Wait, but the vertical leg of the left triangle is \( \sqrt{10^{2}-4^{2}}=\sqrt{84}\approx9.165 \), which is longer than 8, so that can't be. So maybe the left triangle has hypotenuse 10, one leg is the vertical leg, and the other leg is 4. The right triangle has hypotenuse 8, one leg is the vertical leg, and the other leg is \( x \). Wait, no, the hypotenuse of the right triangle can't be shorter than the vertical leg. So I must have misread the diagram. Wait, maybe the left triangle: leg 4, hypotenuse 10, right - angled at the bottom. The right triangle: leg 8, hypotenuse \( x \), and the vertical leg is the same as the vertical leg of the left triangle. Wait, no, the problem says "solve for x", and the right triangle is right - angled at the right, with one leg \( x \), one leg equal to the vertical leg of the left triangle, and hypotenuse 8. Wait, no, let's start over.

Let the vertical segment (the common leg) be \( h \).

For the left right - triangle:
\( h^{2}+4^{2}=10^{2} \)
\( h^{2}=100 - 16 = 84 \)

For the right right - triangle:
\( h^{2}+x^{2}=8^{2} \)
Substitute \( h^{2}=84 \):
\( 84+x^{2}=64 \)
\( x^{2}=64 - 84=-20 \)

This is impossible, which means I misinterpreted the diagram. Wait, maybe the right triangle has hypotenuse 8, and the vertical leg is \( h \), and the horizontal leg is \( x \), but the left triangle has hypotenuse 10, horizontal leg 4, and vertical leg \( h \). Wait, no, maybe the right triangle has legs \( x \) and 8, and hypotenuse equal to the vertical leg of the left triangle? No, the problem says "solve for x", leave in simplest radical form. Wait, maybe the left triangle: \( 10^{2}=4^{2}+h^{2}\), so \( h^{2}=100 - 16 = 84 \). The right triangle: \( 8^{2}=x^{2}+h^{2}\)? No, that gives negative. Wait, maybe the right triangle has hypotenuse 8, and one leg is \( h \), and the other leg is \( x \), but \( h \) is a leg, so \( x^{2}=8^{2}-h^{2} \). But \( h^{2}=84 \), so \( x^{2}=64 - 84=-20 \), which is impossible. So I must have made a mistake in the diagram. Wait, maybe the left triangle has leg 10, hypotenuse 4? No, hypotenuse is longer than leg. Wait, maybe the left triangle: leg 4, leg \( h \), hypotenuse 10. The right triangle: leg \( h \), leg \( x \), hypotenuse 8. But \( 10>8 \), so \( h=\sqrt{10^{2}-4^{2}}=\sqrt{84}\), and \( 8^{2}=x^{2}+h^{2}\) implies \( x^{2}=64 - 84=-20 \), which is impossible. So there must be a mistake in my understanding. Wait, maybe the left triangle is a right triangle with legs 4 and \( h \), hypotenuse 10. The right triangle is a right triangle with legs \( x \) and \( h \), hypotenuse 8. But since \( 10>8 \), \( h=\sqrt{10^{2}-4^{2}}=\sqrt{84}\), and \( 8^{2}=x^{2}+h^{2}\) gives \( x^{2}=64 - 84=-20 \), which is impossible. So maybe the diagram is such that the left triangle has hypotenuse 10, one leg is \( x \), and the other leg is 4, and the right triangle has hypotenuse 8, one leg is \( x \), and the other leg is... No, that doesn't make sense. Wait, maybe the two triangles share the same height \( h \), the left triangle has base 4 and hypotenuse 10, the right triangle has base \( x \) and hypotenuse 8, and both are right - angled at the bottom and right respectively, with the same height \( h \). So for the left triangle: \( h^{2}+4^{2}=10^{2}\Rightarrow h^{2}=84 \). For the right triangle: \( h^{2}+x^{2}=8^{2}\Rightarrow x^{2}=64 - 84=-20 \), which is impossible. So there must be a typo or mis - diagram. Wait, maybe the left triangle has hypotenuse 10, one leg is 8, and the other leg is 4? No, \( 4^{2}+8^{2}=16 + 64 = 80 eq100 \). Wait, maybe the left triangle has leg 6, hypotenuse 10, then \( h = 8 \), but the problem says leg 4. Wait, maybe the right triangle has hypotenuse 10, and the left has hypotenuse 8? No, the problem says left side 10, right side 8. Wait, I think I see the mistake. The right triangle: the hypotenuse is 8, and one leg is the height \( h \), and the other leg is \( x \). The left triangle: hypotenuse 10, one leg is 4, and the other leg is \( h \). But \( h=\sqrt{10^{2}-4^{2}}=\sqrt{84}\), and then in the right triangle, \( x^{2}+h^{2}=8^{2}\) is wrong. Wait, no, the right triangle: the leg is \( x \), the other leg is \( h \), and the hypotenuse is 8? No, the hypotenuse should be the longest side. So \( h=\sqrt{84}\approx9.165>8 \), so the hypotenuse of the right triangle can't be 8. So maybe the right triangle has hypotenuse \( h \), and legs 8 and \( x \), and the left triangle has hypotenuse 10, legs 4 and \( h \). Let's try that.

For the left triangle: \( 4^{2}+h^{2}=10^{2}\Rightarrow h^{2}=100 - 16 = 84\Rightarrow h=\sqrt{84} \)

For the right triangle: \( 8^{2}+x^{2}=h^{2}\Rightarrow64 + x^{2}=84\Rightarrow x^{2}=84 - 64 = 20\Rightarrow x=\sqrt{20}=2\sqrt{5} \)

Ah! That makes sense. I had the Pythagorean theorem reversed for the right triangle. The right triangle has legs 8 and \( x \), and hypotenuse \( h \) (the same as the leg of the left triangle). So the left triangle: legs 4 and \( h \), hypotenuse 10. The right triangle: legs 8 and \( x \), hypotenuse \( h \). That way, \( h \) is the hypotenuse of the right triangle, so \( h>8 \), which is true because \( h=\sqrt{84}\approx9.165>8 \).

Step1: Find \( h^{2} \) from the left triangle

Using the Pythagorean theorem for the left right - triangle with legs 4 and \( h \), and hypotenuse 10:
\( 4^{2}+h^{2}=10^{2} \)
\( 16 + h^{2}=100 \)
Subtract 16 from both sides: \( h^{2}=100 - 16 = 84 \)

Step2: Find \( x \) from the right triangle

Using the Pythagorean theorem for the right right - triangle with legs 8 and \( x \), and hypotenuse \( h \) (where \( h^{2}=84 \)):
\( 8^{2}+x^{2}=h^{2} \)
Substitute \( h^{2}=84 \):
\( 64+x^{2}=84 \)
Subtract 64 from both sides: \( x^{2}=84 - 64 = 20 \)
Take the square root of both sides: \( x=\sqrt{20}=2\sqrt{5} \)

Answer:

\( 2\sqrt{5} \)