A circle has center at origin and radius 1. The points X, Y, and Z lie on the circle such that the length of arc XYZ is (2/3)Ï€.What is the length of line segment XZ?

A. √2
B. √3
C. 2
D. 2+√3
E. 2+√2

The OA is the option B.

How can I solve this PS question? I don't know how to find the length of XZ? Experts, could you help me? Thanks.

Hi Vincen,
Let's take a look at your question.

Center of the circle is (0, 0).
Radius = 1



We will use the angle XOZ first using the arc length formula.
$$S=\ r\left(theta\right)$$ $$\frac{2\pi}{3}=1\left(theta\right)$$
$$theta=\frac{2\pi}{3}$$

Now we can find the length of XZ using cosine formula in triangle XOZ.
The cosine formula is:
$$a^2=b^2+c^2-2bc\ Cos\left(A\right)$$
Where A is the angle opposite to the side length a and between the sides b and c.
$$\left(XZ\right)^2=\left(1\right)^2+\left(1\right)^2-2\left(1\right)\left(1\right)Cos\left(\frac{2\pi}{3}\right)$$
$$\left(XZ\right)^2=1+1-2\left(-\frac{1}{2}\right)$$
$$\left(XZ\right)^2=2+1$$
$$\left(XZ\right)^2=3$$
$$XZ=\sqrt{3}$$

Therefore, option B is correct.

Hope it helps.
I am available if you'd like any follow up.