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Problem image

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

Question: How do I find chord NG?

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    What is NG? The length of the line segment from point N to point G or the length of the curve from point N to point G? – 2017-02-11
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    Please type your question in the post. – 2017-02-11
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    The length of the chord. The length of the arc is just 90 @BrevanEllefsen – 2017-02-11
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    Where does *that* tag come from? – 2017-02-11
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    Whoops... that really has nothing to do with it. I saw it as 'chord' on my browser...@Lonidard – 2017-02-11

2 Answers 2

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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.

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    Victoria- I just read your meta post about being inclusive to all difficulties. You definitely are :) Thanks for answering this trivial question.. – 2017-02-11
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    It might be trivial to you, but way back when you were in Geometry 1 it wasn't. – 2017-02-11
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    Wow, so psychic! Right now, I am in middle school, taking Honors Geometry :) – 2017-02-11
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    Hey, I teach that stuff. I also God help me have to try to read writing from little $%^& (censored). You learn to be psychic. – 2017-02-11
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    Haha, laughing so hard right now – 2017-02-11
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    @IrreverentSalutes Private jokes ? – 2017-02-11
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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 $\overline{NG} = \sqrt{2} \cdot\overline{NO}$

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    Oh noes... I wish I could accept 2 answers!! Thank you for your indepth one!! – 2017-02-11
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    @IrreverentSalutes No problem. You *can* up-vote answers if you want... just hit the giant up-arrows to the left of each answer – 2017-02-11
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    Sadly, I possess less than the magic 125 reputation :( – 2017-02-11
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    @IrreverentSalutes Oh yes, the initial reputation requirements. Oh well, glad I could help at least :) – 2017-02-11