How do I find chord NG?

Question

I have tried extending the lines, creating inscribed angles, and using the Pythagorean theorem.

Question: How do I find chord NG?

Answer 1

First you need the radius ON.

Draw a perpendicular OT to ND.

Since triangle OND is isosceles with ON = OD = radius, you have divided triangle OND into two congruent right triangles with NT = TD = 1/2 (16) = 8.

Use the fact that angle NOD = 1/2(35 degrees) and the inverse sine function to find ON = radius.

Once you have that you have OG and Pythagoras or whatever will give you NG.

You're also asked to find arc GM; just find the angle GOM and work out the fraction of the circumference.

Answer 2

By Law of Cosines we know that $$c^2 = a^2+b^2-2ab\cos(C)$$ Which apply here to find that $$\overline{ND}^2=\overline{NO}^2+\overline{DO}^2-2\overline{NO}\,\overline{DO}\cos(35^o)$$ We now use the fact that $\overline{ND}=16$ and $\overline{NO} = \overline{DO}$ to find that $$16^2=2\overline{NO}^2\bigg(1-\cos(35^o)\bigg)$$
Solve for $\overline{NO}$ and note that triangle $NOG$ is isosceles, and so $\overline{NG} = \sqrt{2} \cdot\overline{NO}$