In a triangle ABC, the bisector of angle A meets base BC at D. Prove that AB>BD?

Question

The triangle is NOT right-angled! I know that people give advise of doing homework when they don't know the answer! The question is not incomplete. This question came in a test I recently wrote and I honestly don't know the answer. I couldn't get it no matter how-much I tried.

The triangle is NOT right-angled! I know that people give advise of doing homework when they don't know the answer! The question is not incomplete. This question came in a test I recently wrote and I honestly don't know the answer. I couldn't get it no matter how-much I tried.

Answer 1

It is a tough Q - coz it is NOT right angle...
The best solution would be to name the angles, LHS Triangle as angle x, y z,,,,, RHS triangle as x', y', z'

The actual proof will be long, but the basic idea is
Sum of the Angles of a Tri = 180
and the angles in straight line = 180

x + y + z =180

x' + y' + z' = 180

z + y' = 180

y' = 180-z

x = x'

x' + 180-z + z' = 180
x + z' = z

Therefore, z is larger Angle than x (or x')

Side opposite to larger angle is larger

Answer 2

Not that hard. Label angle BAC as a, ABC as b, etc.

Look at triangle ABD. It has angles a/2, b, and d, where a/2 + b + d = 180. (d = angle ADB)

Also, a + b + c = 180, so setting these equal we get

a/2 + b + d = a + b + c
d = a/2 + c.

Now the angle opposite side BD in ABD is a/2, and opposite AB is d = a/2 + c. Since c is > 0 this means d > a/2, so by the law of sines BD < AB
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sshanbhag, that is not right, it is certainly possible for BD > AB if it is not bisected. Imagine a triangle with a very short AB and a very long BC. For most D between B and C BD will be longer than AB.

Answer 3

Let me try this answer by using the priciple of assumptions - and using Reductio Ad Absurdum
We are asked to prove is in a traingle ABC when angle A is bisected and meets side BC at D -- AB > BD . Also it is the same as proving that AC > CD as B and C can just be interchanged and question remains same (there are no constrints on B and C for interchanging)
To begin with we say
Let us assume the opposite is true
namely we assume AB< BD and therefore AC < CD by the same logic and assumption (as explained above)
Now combining the two statements mathematically
AB + AC < BD + CD
gives AB + AC < BC ( B,C,D are in one line and B-D-C)
The statement AB + BC < BC is a techincally
incorrect statement becasue if true the traingle cannot be constructed (in all cases the sum of two sides of a traingle is greater than the 3rd side)
Therefore our assumption is false and using Reductio Ad Absurdum we say that AB > BD (proved)

Additional Conclusions and observations
This is true irrespective of whether angle is bisected or not. Only condition is the line dividing the angle should be such that D is between A & C

Answer 4

Because AD is bisector, we have: BD/DC = AB/AC = 26/30 (1) on the other hand BD + DC = BC = 28 (2) (1) and (2) ⇒ BD = 13 and DC = 15 AD is bisector of A, so ∠HAE = ∠DAC = x EG || AC ⇒ ∠DEG = ∠DAC = x ⇒ ∠DEG = ∠AEH = x ⇒ AEH is isosceles ⇒ AH = EH (3) ∠BEH = ∠BEA - ∠HEA = 90 - x ∠BHE = ∠HAE + ∠HEA = x + x = 2x ⇒ ∠EBH = 180 - (90-x + 2x) = 90 - x = ∠BEH ⇒ BEH is isosceles ⇒ BH = EH (4) (3) and (4) ⇒ BH = HA = 13 GH || AC ⇒ BG/BC = BH/BA = 1/2 ⇒ BG = BC/2 = 14 ⇒ DG = BG - BD = 14 - 13 = 1

Answer 5

i think the line AD is meant to be perpendicular to BC. If that's true, then AB > AD because it's the hypotenuse of a right-angled triangle.

Answer 6

i forget how to prove... It's been years since I did that... however the hypotenuse is always the longest side in a right triangle... BA is the hypotenuse opposite the angle BDA... so BD and AD is shorter than BA...

Answer 7

as the A angle a is bisected hence angle ADC greater hence AB is greater since the side opposite to greater side is greater

Answer 8

As such, AB can be smaller than BD. There must be some other condition.

Answer 9

Yeah, what she said. Do your own homework.

Another suggestion however, would be to actually do it, and then if you would like feedback, post your solution asking for feedback. People respect when you took the time to do the work, and will be more willing to help you.

Answer 10

I think your question is incomplete.