What is the proof for triangle ABC with known sides and angle relationships?

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Discussion Overview

The discussion revolves around proving relationships in triangle ABC, specifically focusing on the lengths of sides and angles given certain conditions. Participants explore the application of the angle bisector theorem and its implications for calculating unknown side lengths in the triangle.

Discussion Character

  • Exploratory
  • Technical explanation
  • Mathematical reasoning
  • Homework-related

Main Points Raised

  • One participant describes their struggle to prove relationships within triangle ABC, specifically the unknown length from point B to point D.
  • Another participant suggests using the angle bisector theorem, stating that it divides the opposite side in the same ratio as the adjacent sides.
  • A participant expresses confusion about applying the theorem without assuming right angles and seeks further clarification on the terms "adjacent" and "opposite."
  • Further clarification is provided, paraphrasing the theorem to relate the segments CD and BD to the sides AC and AB.
  • A participant calculates the length BD using the ratios derived from the angle bisector theorem, arriving at a numerical solution.
  • Another participant corrects their calculation and provides a different numerical result for the length BD.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the initial understanding of the angle bisector theorem, but there is agreement on its application to derive the length BD. The final numerical results differ, indicating unresolved calculations.

Contextual Notes

Participants express uncertainty about the application of the angle bisector theorem without additional information, and there are unresolved aspects regarding the assumptions needed for the calculations.

Who May Find This Useful

Students studying geometry, particularly those interested in triangle properties and theorems related to angle bisectors.

Blissfulpain
ok, I'm in first year coolege and we're doing some review from high school work.. but i can't for the life of me remember how to do one of these questions and i can't seem to get past this one part or at least figure out how to prove that it is what i think it is.

it's going to be a bit trickier, because i can't insert a picture, but i guess that isn't of the most importance anyways. alright

you have triangle ABC.
side AB=4 units
side AC=7 units
the line CD extends for 3 units towards point B, the distance between point B an D is unknown and is what needs to be found
the Angle BCD is = angle DAC

here's what I've tried to do so far...

i've tried to fold the 2 triangles in half across the line AD, which lines up the two sides AB and AC, with the line AC extending 3 units futher then line AB. I'm still trying to follow this train of reasoning.. but so far with little success, i don't really know what to do next, but i think it helps somehow : )

next...

I tried extending the common side between the two triangles, line AD, far enough so that i can draw a new line, 7 units long, from point C, i'll call the intersection point E. The problem arises when i can't prove that the new triangle DEC is the same as triangle ADC. I CAN prove that it is proportional to triangle ABD, but that doesn't help me, because i only know that the side CE is 7 units, the Side DC is 3 units, and i don't know if i can prove that the angle DEC is equal to the angle DAC... which means i can't prove that the line DE is a continuation of the line AD...

hence I'm lost.

thx for the help in advance.. hey, kewl... just found the attacthment thing... i'll go draw a picture and then post :)
 

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Hint: angle bisector theorem.

Hope that helps!
 
nope, doesn't help... i looked it up on the net, and got this "The angle bisector of an angle in a triangle divides the opposite side in the same ratio as the sides adjacent to the angle."

so the angle bisector is line AD, and the adjacent and opposiye side are? i can't assume ANY right angles... and all the other stuff i found on this theorem had something to do with area of a triangle, which i can't do either, because i don't have enough info to find that yet...

a bit more of a hint is needed :confused:
 
Blissfulpain said:
nope, doesn't help... i looked it up on the net, and got this "The angle bisector of an angle in a triangle divides the opposite side in the same ratio as the sides adjacent to the angle."

so the angle bisector is line AD, and the adjacent and opposiye side are? i can't assume ANY right angles... and all the other stuff i found on this theorem had something to do with area of a triangle, which i can't do either, because i don't have enough info to find that yet...

a bit more of a hint is needed :confused:

No right angles required, merely an understanding of the words adjacent and opposite.

Let me paraphrase the theorem for the triangle you've got :

Simply, it says that CD/BD = AC/AB.

Now go back and understand what the theorem is trying to say.
 
so, 3/x=4/7
3/x=1.75
x=1.75*3
x=5.25
huh, i had gotten that number before... just couldn't prove that it was the right number... now i can with that theorem

thx a lot guys :)))))
 
oops... i mean 3/7=x/4
so 3/7=.429
.429x4=1.7 units

therefore BD=1.7
 

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