Tough Geometry problem

Question

Please help with this one :

A right angled triangle ABC was given in which angle C is 90 degrees. BC=4, AD=3. D is a point on BC. Given that the perimeter of triangle ABD is equal to the perimeter of triangle ADC. Then find the ratio BD/DC.

chaoswithin · Answer

Since you posted this on PS forum, could you give the answer choices as well?

chaoswithin · Answer

The reason why I ask is because I can get a range for the possible answer, but not an exact number. To get the upper limit, we can use the fact that the perimeters of the two triangles are equal and since they share the side AD, we can see that AB+BD = AC + DC and since AB has to be > AC, BD has to be < DC so BD/DC has to be < 1. for the lower limit, since AD = 3, DC has to be < 3. and since BD + DC = 4, when DC = 3 BD = 1 and BD/DC has to be > 1/3. Therefore, 1/3 < BD/DC < 1. This is the best I can do with the givens. If anyone else could shine some light on this problem, I would appreciate it.

subhashghosh · Answer

The answer given was 1:3, I could not reckon how though !

chaoswithin · Answer

This answer doesn't make sense.

Using the givens, the triangle should look similar to the triangle in the attachment.

In order for BD/DC to be 1/3, BD = 1 and DC = 3.

But since triangle ADC has hypotenuse AD = 3. DC cannot equal 3.

In order for BD/DC to be 1/3, the original question should be rephrased so  AC = 3  instead of  AD = 3 .

If AC = 3.

AB = 5 (b/c the triangle becomes 345 triangle).

and we are left with two equations and two unknowns where

AB + BD = AC + DC and BD + DC = 4.

which results in BD = 1 and DC = 3.

shtrini · Answer

I thought this problem was really hard, but actually it's really really simple.

We know that triangle ABC is a 3-4-5 triangle. simple from knowing AC = 3 and CB = 4

CB = 4 = CD + DB

Equation 1 : CD + DB = 4

We are told that P(ABD) = P(ADC)

3 + CD + AD = 5 + DB + AD

Equation 2 : CD - DB = 2

CD + DB = 4
 CD - DB = 2 
2CD = 6

CD = 3

Therefore DB = 1

And finally CD:DB > 3:1 or whatever direction the question asked.

Essentially you don't need to know AD.

bumpbot · Answer

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Tough Geometry problem

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