Calculate the angle between the lines

Question

I need to calculate the angle x. I feel like I'm missing something basic since this is supposed to be an easy exercise but I can't seem to get it. It's under the chapter trigonometry.

AB is 8 units long, the radius is 5. I was looking for right angle triangles, but didn't find a way to make use of the info.

Note that SA=SB=5, because that’s the radius – Divide1918 Dec 11 '23 at 14:30 — Divide1918, Dec 11 '23 at 14:30
First find RB. Then look at triangle RSB. – Saeed Dec 11 '23 at 17:03 — Saeed, Dec 11 '23 at 17:03

John Omielan · Accepted Answer · 2023-12-11T16:42:51.693

You're correct to look for right-angled triangles. In particular, as shown below in a modified version of your diagram, the midpoint of $AB$ is labeled as $C$, and with the line segments of $SA$, $SC$ and $SB$ added.

As Divide1918's comment indicates, $\lvert SA\rvert = \lvert SB\rvert = 5$ as they are the radii. Thus, $\triangle ASB$ is isosceles, so with $C$ being the midpoint of $AB$, then $\measuredangle SCB = 90^{\circ}$. Since $\lvert CB\rvert = 4$, then using the Pythagorean theorem with $\triangle SBC$, we get that

$$\lvert SC\rvert = \sqrt{5^2 - 4^2} = \sqrt{9} = 3$$

With $\triangle SCR$ being right-angled, we have

$$\sin(x) = \frac{\lvert SC\rvert}{\lvert SR\rvert} = \frac{3}{12} = 0.25 \;\;\;\to\;\;\; x = \sin^{-1}(0.25)$$

Thanks guys, I forgot that the height of the SAB triangle cuts AB in half. — Mixoftwo, Dec 11 '23 at 21:37

Calculate the angle between the lines

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