If angle BAD is a right angle, what is the length of side BD

[jennifer.f.rizk]

If angle BAD is a right angle, what is the length of side BD?

(1) AC is perpendicular to BD

(2) BC = CD

In this question why is triangle BAD isoscles?-This is from the geometry question bank.

[Ben Ku]

If angle BAD is a right angle, what is the length of side BD?

(1) AC is perpendicular to BD

(2) BC = CD

In this question why is triangle BAD isoscles?-This is from the geometry question bank.

In this problem, you can infer BAD is isosceles ONLY from statement (2). Basically, AC splits the right triangle into two smaller triangles (ACB and ACD). Now these two triangles are congruent to each other, since they both share AC as one leg, and the other legs have equal lengths (BC=CD).

Since these two smaller triangles (ACB and ACD) are congruent, then their hypotenuses also have the same length (AB=AD). Therefore triangle BAD is an isosceles right triangle.

[carole.okigbo]

great, thanks for the explaination

[Ben Ku]

You're welcome!

[zdgura]

The correct answer to this is C. Ben your logic (and I apologize if I'm incorrect here) seems to make me think that B would suffice as an answer. Is C the correct answer because while stem 2 lets you know that you have two sides that are the same you don't know whether you have any angles that are the same. Stem 1 lets you know that the two are right triangles and you know have two sides being equal and you have angle information as well....is that correct?

[tim]

You're right. 2 alone is not sufficient to conclude that the triangle is isosceles. You need both statements..

[lj6871849]

Hi Tim - one petite doubt.

Since AC is perpendicular to BD do we know the angle split at A is 45-45?

or is it because AC is perpendicular bisector do we know the angles formed at A by the like AC is 45-45?

Basically what i mean to ask is if we know AC is perpendicular to BD then do we know angle at is 45-45?
OR
If we know AC is perpendicular bisector of BD then do we know the angles formed at right angle A by AC is 45 degrees each?

Cheers

[jlucero]

The second one. Otherwise this couldn't happen:

[SoumyaA472]

I came across this question in the Geometry question bank, so thought it is easier to post on an existing thread.
I was wondering why the 3 triangles are not similar based on A alone? Angle BAD = BCA = ACD = 90 degrees.

And based on this information and the side AD= 5 (given) , we should be able to determine the length of BD.

[tim]

Can you post a screen shot of the relevant diagram so other students will be able to benefit from our answer? I see there was supposed to be an attachment in a previous post, but I can't access it.

[AliasgharA585]

Hi
I just wanted to know if statement 1 would be sufficient since if AC is the perpendicular bisector then we know that it is 45-45-90 triangle and one side AD is known.From that we can calculate AC and similarly in triangle ACD which is a 45-45-90 triangle and side AC is known and can calculate side BC?

[RonPurewal]

can you please post a picture of the diagram? thanks.

[RonPurewal]

in any case-- if BAD is a RANDOM right triangle, then AC (in statement 1) is not necessarily a perpendicular BISECTOR.

to see this, just draw a triangle BAD that is decidedly NOT isosceles—e.g., draw a 10º-80º-90º triangle (or suchlike).
then draw the perpendicular.
you'll notice that it's not even close to being a bisector. (if 'D' is the smaller angle, then CD is going to be WAY longer than CA.)

[AliasgharA585]

Thanks Ron for the explanation...

[tim]

Glad it helped!