## anonymous one year ago Find the missing measures.

1. anonymous

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2. mathstudent55

What do you know about triangles IFE and IHG?

3. anonymous

I know that HI=$3\sqrt{3}$

4. mathstudent55

That is correct, but what about my question?

5. anonymous

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

6. mathstudent55

No, that's not what I mean. Look at the triangles when you separate them. See the figure below. |dw:1437458809533:dw|

7. nincompoop

understand the concept of proportionality or similarity and congruence.

8. 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

9. 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.

10. nincompoop

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11. 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.

12. mathstudent55

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

13. 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.

14. mathstudent55

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15. mathstudent55

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

16. mathstudent55

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17. mathstudent55

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

18. nincompoop

basic principle: |dw:1437459224575:dw|

19. nincompoop

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20. nincompoop

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21. mathstudent55

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

22. triciaal

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23. mathstudent55

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24. mathstudent55

We just need to find FH.

25. mathstudent55

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

26. nincompoop

use tangent ratio

27. nincompoop
28. 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.

29. mathstudent55

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

30. nincompoop

@triciaal |dw:1437460115571:dw|