Find the missing measures.

- anonymous

Find the missing measures.

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- anonymous

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- mathstudent55

What do you know about triangles IFE and IHG?

- anonymous

I know that HI=\[3\sqrt{3}\]

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## More answers

- mathstudent55

That is correct, but what about my question?

- anonymous

IFG has 2 side missing and IHG has 1 missing side?

- mathstudent55

No, that's not what I mean.
Look at the triangles when you separate them.
See the figure below.
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- nincompoop

understand the concept of proportionality or similarity and congruence.

- nincompoop

if you want, you can solve the smaller portion of right triangle's adjacent side (adjacent to the angle), and then work your way from there

- mathstudent55

Now look.
Triangle IFE and triangle IHG are both right triangles.
That means that each one has a right angle.
Right angles are congruent, so you already have one pair of congruent angles.

- nincompoop

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- mathstudent55

Now also notice that angle I of triangle IFE and angle I of triangle IHG are congruent because they are the same angle.
That is a second pair of congruent angles.
That makes the triangles similar.

- mathstudent55

@nincompoop If instead of writing, you did some more reading, you'd see he already found that out.

- mathstudent55

Since we now know we have similar triangles, we use the fact that the corresponding sides have proportional lengths. We set up proportions and find the missing lengths.

- mathstudent55

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- mathstudent55

We can use the two sides in black above to establish a ratio of lengths:
\(\dfrac{large~triangle}{small~triangle} = \dfrac{5}{3} \)

- mathstudent55

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- mathstudent55

\(\dfrac{5}{3} = \dfrac{EG + 6}{6} \)
Do you understand this proportion?

- nincompoop

basic principle:
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- nincompoop

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- nincompoop

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- mathstudent55

\(\dfrac{5}{3} = \dfrac{EG + 6}{6}\)
\(6 \times 5 = 3 \times (EG + 6) \)
\(30 = 3EG + 18\)
\(3 EG = 12\)
\(EG = 4\)

- triciaal

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- mathstudent55

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- mathstudent55

We just need to find FH.

- mathstudent55

\(\dfrac{6}{3\sqrt 3} = \dfrac{4}{FH} \)
\(6FH = 4 \times 3 \sqrt 3\)
\(6 FH = 12 \sqrt 3\)
\(FH = 2 \sqrt 3\)

- nincompoop

use tangent ratio

- nincompoop

page 41
http://www.scribd.com/doc/22374733/Mathematics-Teach-Yourself-Trigonometry

- mathstudent55

We can find FH another way.
We can solve for FI using the Pythagorean theorem with the large triangle, then subtract HI from it.
\((FI)^2 + (FE)^2 = (EI)^2\)
\((FI)^2 + 5^2 = 10^2\)
\((FI)^2 + 25 = 100\)
\((FI)^2 = 75\)
\(FI = 5\sqrt3\)
\(FH = FI - HI\)
\(FH = 5 \sqrt 3 - 3\sqrt 3\)
\(FH = 2 \sqrt 3\)
As you can see, we get the same result for FH as we did above.

- mathstudent55

@SimiYami
I hope you can follow my explanations above.
If you have any questions, just ask.

- nincompoop

@triciaal |dw:1437460115571:dw|

- anonymous

I understood what you've explained. Thank you very much for your help again @mathstudent55. Thank you @nincompoop for your example.:)

- triciaal

@nincompoop trig not necessary here
I would use pythagorean theorem as shown above
even easy to do ratio shown above by @mathstudent55

- mathstudent55

@SimiYami You're welcome.

- triciaal

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