Question

find the value of x to the nearest tenth.

Explanation:

Step1: Find the height of the right triangle (shared side)

In the smaller right triangle with legs 3 and 6, the hypotenuse (which is the height of the larger right triangle) can be found using the Pythagorean theorem? Wait, no, actually, the two right triangles share a common leg. Wait, the smaller triangle has legs 3 and 6? Wait, no, the smaller triangle has legs 3 and the vertical side, and hypotenuse 6? Wait, no, looking at the diagram: the smaller right triangle has legs 3 (horizontal) and let's call the vertical leg \( h \), and hypotenuse 6? Wait, no, maybe the smaller triangle has legs 3 and \( h \), and the other triangle has hypotenuse 10 and legs \( x \) and \( h \). Wait, let's re-examine.

Wait, the diagram shows two right triangles: one with legs \( x \) and \( h \) (hypotenuse 10), and another with legs 3 and \( h \) (hypotenuse 6)? Wait, no, the smaller triangle has legs 3 and \( h \), and hypotenuse 6? Wait, no, maybe the smaller triangle is a right triangle with legs 3 and 6? Wait, no, the right angle is at the bottom right of the smaller triangle, so legs are 3 (horizontal) and \( h \) (vertical), and hypotenuse 6. Wait, no, the hypotenuse of the smaller triangle is 6, one leg is 3, so we can find \( h \) using Pythagorean theorem: \( h^2 + 3^2 = 6^2 \)? Wait, no, that would be \( h^2 + 9 = 36 \), so \( h^2 = 27 \), \( h = \sqrt{27} = 3\sqrt{3} \approx 5.196 \). Wait, but then the larger triangle has hypotenuse 10, leg \( h \), so \( x^2 + h^2 = 10^2 \), so \( x^2 = 100 - h^2 = 100 - 27 = 73 \), so \( x = \sqrt{73} \approx 8.5 \). Wait, but let's check again.

Wait, maybe the smaller triangle has legs 3 and 6, and the larger triangle shares the vertical leg with the smaller one. Wait, no, the right angle in the smaller triangle is at the bottom, so horizontal leg 3, vertical leg \( h \), hypotenuse 6. Then the larger triangle has horizontal leg \( x \), vertical leg \( h \), hypotenuse 10. So first, find \( h \) from the smaller triangle: \( h = \sqrt{6^2 - 3^2} = \sqrt{36 - 9} = \sqrt{27} = 3\sqrt{3} \approx 5.196 \). Then, in the larger triangle, \( x = \sqrt{10^2 - h^2} = \sqrt{100 - 27} = \sqrt{73} \approx 8.5 \). Wait, but let's confirm.

Alternatively, maybe the smaller triangle is a right triangle with legs 3 and 6, and the larger triangle is similar? No, not necessarily. Wait, the key is that the two right triangles share the vertical leg. So first, calculate the vertical leg (height) using the smaller triangle: legs 3 and 6? Wait, no, 3 and \( h \), hypotenuse 6. So \( h = \sqrt{6^2 - 3^2} = \sqrt{27} \approx 5.196 \). Then, in the larger triangle, hypotenuse 10, leg \( h \), so \( x = \sqrt{10^2 - h^2} = \sqrt{100 - 27} = \sqrt{73} \approx 8.5 \). So to the nearest tenth, 8.5.

Step1: Calculate the shared vertical leg (h)

In the smaller right triangle with legs 3 and hypotenuse 6, use Pythagorean theorem:
\( h^2 + 3^2 = 6^2 \)
\( h^2 = 36 - 9 = 27 \)
\( h = \sqrt{27} \approx 5.196 \)

Step2: Calculate x using the larger right triangle

In the larger right triangle with hypotenuse 10 and leg \( h \), use Pythagorean theorem:
\( x^2 + h^2 = 10^2 \)
Substitute \( h^2 = 27 \):
\( x^2 = 100 - 27 = 73 \)
\( x = \sqrt{73} \approx 8.5 \)

Answer:

\( \boxed{8.5} \)