Indirect Geometric Proof

greatwhiteshark

Full Member
Joined
May 8, 2005
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279
Given:

Triangle ABC is scalene.
Segment BD bisects angle ABC.

Prove:

Segment BD is NOT perpendicular to segment AC.

I am totally lost. Help.
 
Hello, greatwhiteshark!

In an Indirect Proof, we work 'backwards'.
We assume the <u>opposite</u> of the punchline
. . and hope to get a contradiction of the given.

Given: Triangle ABC is scalene.
Segment BD bisects angle ABC.

Prove: Segment BD is NOT perpendicular to segment AC.
By definition, a scalene triangle has three <u>unequal</u> sides and angles.

We assume that BD <u>is</u> perpendicular to AC.

1. <u>/</u>BDA = <u>/</u>BDC = 90<sup>o</sup> . . . . . . . 1. definition of perpendicular

2. <u>/</u>ABD = <u>/</u>CBD. . . . . . . . . . . . .2. Given (BD bisects <u>/</u>ABC)

3. BD = BD . . . . . . . . . . . . . . . 3. Identity

4. Δ BDA congruent Δ BDC . . . 4. a.s.a.

5. AB = BC . . . . . . . . . . . . . . . 5. 'corresponding parts'


But this means that triangle ABC is <u>isosceles</u> (two equal sides),
. . and we know that the triangle is scalene (<u>no</u> equal sides).

We have reached a contradiction.
. . Hence, our assumption was incorrect.

Therefore, BD is not perpendicular to AC.
 
okay

I better practice more direct proofs before stepping into the waters of indirect proofs. I like this stuff, really. My teacher said that we will be given such proofs in several August home exams.
 
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