Question

question 10
10 pts
g.gsr.6.3 (hc)
find the value of x.
image of a triangle with base 35 + 8, hypotenuse 37, height as red dashed line, and side x
accessible description
25
√208
√125
√325

Explanation:

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

Using the Pythagorean theorem for the left triangle with hypotenuse 37 and base 35: \( h^2 + 35^2 = 37^2 \)
\( h^2 = 37^2 - 35^2 = (37 - 35)(37 + 35) = 2 \times 72 = 144 \)
So \( h = 12 \) (since height is positive)

Step2: Find x using the right triangle with base 8 and height 12

Using the Pythagorean theorem for the right triangle: \( x^2 + 8^2 = h^2 \) (wait, no, actually the right triangle has legs 8 and h, hypotenuse x? Wait, no, the red dashed line is the height, so the right triangle on the right has legs 8 and h, hypotenuse x. Wait, we found h = 12, so:
\( x^2 = 12^2 + 8^2 = 144 + 64 = 208 \)? Wait, no, wait, maybe I mixed up. Wait, the left triangle: hypotenuse 37, base 35, so height h: \( h = \sqrt{37^2 - 35^2} = \sqrt{(37 - 35)(37 + 35)} = \sqrt{2 \times 72} = \sqrt{144} = 12 \). Then the right triangle: legs are 8 and h (12), so hypotenuse x: \( x = \sqrt{12^2 + 8^2} = \sqrt{144 + 64} = \sqrt{208} \)? Wait, but let's check the options. Wait, maybe I made a mistake. Wait, the total base is 35 + 8 = 43? No, the figure shows the base is split into 35 and 8, so the left triangle has base 35, hypotenuse 37, the right triangle has base 8, and the height is the same. So height h: \( h = \sqrt{37^2 - 35^2} = 12 \), then x is the hypotenuse of the right triangle with legs 8 and h (12), so \( x = \sqrt{8^2 + 12^2} = \sqrt{64 + 144} = \sqrt{208} \). Wait, but let's check the options. Option B is \( \sqrt{208} \), but wait, maybe I messed up. Wait, no, let's recalculate: 37 squared is 1369, 35 squared is 1225, 1369 - 1225 = 144, so h is 12. Then the right triangle: 8 squared is 64, 12 squared is 144, 64 + 144 = 208, so x is \( \sqrt{208} \). Wait, but let's check the options. The options are 25, \( \sqrt{208} \), \( \sqrt{125} \), \( \sqrt{325} \). Wait, maybe I made a mistake in the triangle. Wait, maybe the red dashed line is the height, so the left triangle is a right triangle with hypotenuse 37, one leg 35, so the other leg (height) is 12. Then the right triangle is a right triangle with one leg 8, the other leg 12, so hypotenuse x: \( x = \sqrt{12^2 + 8^2} = \sqrt{208} \), which is option B. Wait, but let's check again. 37^2 = 1369, 35^2 = 1225, 1369 - 1225 = 144, so height is 12. Then 12^2 + 8^2 = 144 + 64 = 208, so x = sqrt(208). So the answer should be \( \sqrt{208} \).

Answer:

\(\sqrt{208}\) (Option: \(\sqrt{208}\))