Question

In the quadrilateral ABCD AB = AC = AD = 12.5, BC = 7 and CD = 15.
Find BD

Answer 1

Could you show us what you've done so far with the question?

Answer 2

Well, i figured out , by using the cosine theorem ( that we're not supposed to use in this problem ) that the angle ACD=ADC = aprox 53° and the angle ACB=ABC = aprox 74°

But that seems uselees since i dont know ABD.

Answer 3

Well first why don't you write down the lengths you know, next to their corresponding sides?

Answer 4

And we will continue from there.

Answer 5

You can use the drawing tool right at the bottom of this reply section.

Answer 6

Well we can start by proving similar triangles.

Answer 7

ABD and ACD would be ideal but I don't think you can prove the two are similar. We can start off small by proving these two triangles shaded.

Answer 8

You can add an extra letter in the middle ie. O

Answer 9

Add as in "label" I mean.

Answer 10

I think it's impossible to prove that ABO is similar or congruent to ADO ( O is the intersection of AC and BD) because we have two sides and one angle equals, but BO is clearly not equal to OD, since this is not a paralelogram.

Answer 11

Nope. We can prove that angle ABD is equal to angle ADb becaue they are base angles of the isosceles triangle ABD.

Answer 12

and then we have a common side AO. And AB=AD.

Answer 13

That's SAS.

Answer 14

Wouldn't be SAS only if angles BAO = DAO?
And if BO = OD, that means that the diagonals would bisect each other , and that is only possible if ABCD was a parallelogram.
And since 7 is not equal to 12.5 , ABCD is not a parallelogram, and is impossible for their diagonals to bisect each other.

Isn't that a contradiction

Answer 15

AB=AD in triangle ADB

Answer 16

THat makes ADB an isosceles triangle.

Answer 17

Yes

Answer 18

So therefore AngleADB=AngleABD. That's one reason why ABO is similar to ADO.

Answer 19

Second reason is that they have a common side AO.

Answer 20

And finally AB=AD. The two triangles can be similar or congruent whichever way you want it to be.

Answer 21

Ok, they are similar, so how we use that?