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# Trigonometry

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Just a question I don't understand, anything will help.

Given a triangle DEF with a segment GH parallel to EF. Find the length of FG given the following information. DG=x-4, FG=2x, DH=x-7, HE=x+3. Point H is located somewhere on line segment ED.

GobroSir  Feb 14, 2018
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Here's  a rough pic of the situation

D

x - 7                    x - 4

H______________G

x + 3                                   2x

E  ______________________________F

A segment parallel to a base of a triangle splits the sides in equal proportions

So

DH / HE  =  DG / GF      so we have

[ x - 7 ] / [ x + 3 ]  = [ x - 4 ] / [ 2x  ]   cross-multiply

2x (x - 7)  =  (x + 3) ( x - 4)   simplify

2x^2  - 14x    =  x^2  - x  - 12       rearrange as

x^2 - 13x  + 12  =  0

Factor   as

(x - 12) ( x - 1)  = 0

Setting to 0  and solving for x produces  x  = 12   or x  = 1

Reject the second soution since it makes (x - 7)  and (x - 4)  negative lengths

So  x  = 12    ⇒   FG  =  2(12)  =  24

Check that this works    5 / 15   =  8 / 24      true  !!!

CPhill  Feb 14, 2018
edited by CPhill  Feb 14, 2018

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