Question

find the value of x to the nearest tenth.
(there is a geometric figure with a right triangle and another right triangle. the hypotenuse of the first right triangle is 10, one leg is x, and the other right triangle has legs 6 and 3.)
answer attempt 1 out of 2
x ≈ |
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Explanation:

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

In the smaller right triangle with legs 3 and 6, we can find the height (let's call it \( h \)) using the Pythagorean theorem? Wait, no, actually, the smaller triangle has legs 3 and 6? Wait, no, the smaller right triangle has legs 3 and the height, and hypotenuse 6? Wait, no, looking at the diagram, the smaller right triangle has one leg 3, hypotenuse 6? Wait, no, the smaller right triangle: one leg is 3, the other leg is the height (let's say \( h \)), and the hypotenuse is 6? Wait, no, maybe I got it wrong. Wait, the two right triangles share a common leg (the height). Let's denote the common leg as \( h \). In the smaller right triangle, the legs are 3 and \( h \), and hypotenuse 6? Wait, no, the smaller right triangle has a right angle, one leg 3, another leg \( h \), and hypotenuse 6? Wait, no, that can't be, because 3-6- \( h \) would mean \( h = \sqrt{6^2 - 3^2} \). Let's calculate that.

So, for the smaller right triangle: \( h = \sqrt{6^2 - 3^2} = \sqrt{36 - 9} = \sqrt{27} = 3\sqrt{3} \approx 5.196 \). Wait, but maybe the smaller triangle has legs 3 and 6? No, the diagram shows the smaller triangle with one leg 3 (horizontal) and hypotenuse 6? Wait, no, the right angle is at the bottom right, so the legs are 3 (horizontal) and \( h \) (vertical), and hypotenuse 6 (the side connecting the two triangles). So yes, \( h = \sqrt{6^2 - 3^2} = \sqrt{27} \approx 5.196 \).

Step2: Find \( x \) using the Pythagorean theorem in the larger right triangle

The larger right triangle has hypotenuse 10 and one leg \( h \) (the height we just found), and the other leg \( x \). So by Pythagorean theorem: \( x = \sqrt{10^2 - h^2} \). We know \( h^2 = 27 \), so \( x = \sqrt{100 - 27} = \sqrt{73} \approx 8.5 \) (to the nearest tenth). Wait, let's check:

First, calculate \( h \): in the smaller right triangle, legs 3 and \( h \), hypotenuse 6? Wait, no, that would mean \( 3^2 + h^2 = 6^2 \), so \( h^2 = 36 - 9 = 27 \), so \( h = \sqrt{27} \approx 5.196 \). Then, in the larger right triangle, hypotenuse 10, one leg \( h \), so the other leg \( x = \sqrt{10^2 - h^2} = \sqrt{100 - 27} = \sqrt{73} \approx 8.544 \), which to the nearest tenth is 8.5.

Wait, but maybe I mixed up the triangles. Let's re-examine the diagram. The two right triangles: the larger one has hypotenuse 10, one leg \( x \) (horizontal), and the other leg \( h \) (vertical). The smaller one has hypotenuse 6, one leg 3 (horizontal), and the other leg \( h \) (vertical). So yes, the vertical leg \( h \) is common. So first, find \( h \) from the smaller triangle: \( h = \sqrt{6^2 - 3^2} = \sqrt{27} \approx 5.196 \). 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 has legs 3 and 6, and hypotenuse \( h \)? No, the right angle is at the bottom right, so the legs are 3 (horizontal) and 6 (vertical)? Wait, that would make the hypotenuse \( \sqrt{3^2 + 6^2} = \sqrt{9 + 36} = \sqrt{45} = 3\sqrt{5} \approx 6.708 \), but the diagram shows the hypotenuse as 6? No, maybe I misread the diagram. Wait, the user's diagram: the smaller right triangle has a right angle at the bottom right, one leg 3 (horizontal), another leg 6 (vertical)? Wait, no, the label is 6 on the hypotenuse? Wait, the diagram shows the smaller triangle with one leg 3 (horizontal), hypotenuse 6 (the side connecting to the larger triangle), and the other leg (vertical) is \( h \). So the right angle is at the bottom right, so legs are 3 (horizontal) and \( h \) (vertical), hypotenuse 6 (the side going up to the larger triangle). So then \( h = \sqrt{6^2 - 3^2} = \sqrt{27} \approx 5.196 \). Then the larger triangle has hypotenuse 10, one leg \( h \), so the other leg \( x = \sqrt{10^2 - h^2} = \sqrt{100 - 27} = \sqrt{73} \approx 8.5 \).

Yes, that makes sense. So:

Step1: Calculate the common height \( h \)

In the smaller right triangle:
\( h = \sqrt{6^2 - 3^2} = \sqrt{36 - 9} = \sqrt{27} \approx 5.196 \)

Step2: Calculate \( x \) using Pythagorean theorem in the larger triangle

In the larger right triangle, hypotenuse = 10, one leg = \( h \), so:
\( x = \sqrt{10^2 - h^2} = \sqrt{100 - 27} = \sqrt{73} \approx 8.5 \) (to the nearest tenth)

Answer:

\( x \approx 8.5 \)