Find the missing measures.

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Find the missing measures.

Mathematics
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|dw:1437458115319:dw|
What do you know about triangles IFE and IHG?
I know that HI=\[3\sqrt{3}\]

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That is correct, but what about my question?
IFG has 2 side missing and IHG has 1 missing side?
No, that's not what I mean. Look at the triangles when you separate them. See the figure below. |dw:1437458809533:dw|
understand the concept of proportionality or similarity and congruence.
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
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.
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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.
@nincompoop If instead of writing, you did some more reading, you'd see he already found that out.
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.
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We can use the two sides in black above to establish a ratio of lengths: \(\dfrac{large~triangle}{small~triangle} = \dfrac{5}{3} \)
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\(\dfrac{5}{3} = \dfrac{EG + 6}{6} \) Do you understand this proportion?
basic principle: |dw:1437459224575:dw|
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\(\dfrac{5}{3} = \dfrac{EG + 6}{6}\) \(6 \times 5 = 3 \times (EG + 6) \) \(30 = 3EG + 18\) \(3 EG = 12\) \(EG = 4\)
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We just need to find FH.
\(\dfrac{6}{3\sqrt 3} = \dfrac{4}{FH} \) \(6FH = 4 \times 3 \sqrt 3\) \(6 FH = 12 \sqrt 3\) \(FH = 2 \sqrt 3\)
use tangent ratio
page 41 http://www.scribd.com/doc/22374733/Mathematics-Teach-Yourself-Trigonometry
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.
@SimiYami I hope you can follow my explanations above. If you have any questions, just ask.
@triciaal |dw:1437460115571:dw|
I understood what you've explained. Thank you very much for your help again @mathstudent55. Thank you @nincompoop for your example.:)
@nincompoop trig not necessary here I would use pythagorean theorem as shown above even easy to do ratio shown above by @mathstudent55
@SimiYami You're welcome.
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