If AC=12, does ACB=90?

Question

https://gmatclub.com/forum/download/file.php?id=40413 If AC = 12, does ACB = 90? (1) BC = 5 (2) AC=CD Capture1.JPG

amanvermagmat · Answer

Statement 1 :
With AC=12 and BC=5, but without knowing the length of AB, we cannot conclude whether angle ACB is 90 or not. IF AB=13, then the triangle will have the sides as (5,12,13) which is a pythagorean triplet, and it will become a right angled triangle. But without that this statement is  not sufficient .

Statement 2 :
IF AC=CD, this means BC is a median. But that doesn't help us in figuring out angle ACB.  Not sufficient .

Combining the statements:
AC=CD=12 and BC which is median is 5. But without knowing anything about sides AB/BD we cannot conclude what angle ACB will be. Eg, if it was given that triangle is isosceles with AB=BD then we would have concluded that angle ACB=90 (because in that case median would be perpendicular to the opposite side). But since that also is not given, even after combining the statements the data is  not sufficient .

Hence  E answer

siddharth19 · Answer

amanvermagmat , hi
in stat(2) , since AC=CD , this also implies that ABC(angle)=DBC(angle) , angle opposite to equal side are equal , so therefore BD is median as well as angle bisector and therefefore also the altitude https://gmatclub.com/forum/medians-alti ... 95489.html
so it is sufficient

please discuss your views Bunuel VeritasKarishma

IanStewart · Answer

Using both statements, all we know is that we have a line 24 units long, and sticking out of the middle of it there's a line that's 5 units long. There's no way to tell what angle they meet at, so the answer is E. The fact that there's a triangle in the diagram might lead to confusion, but if you erase the sides AB and BD, it might be easier to see why we don't have enough information.

I'd add that there's a danger in memorizing rules and formulas in geometry, and trying to apply them in situations where they possibly don't work, rather than thinking about how much flexibility the information in each statement gives you to draw the diagram in different ways. The facts cited in the post above about bisectors are true for isosceles triangles, but we don't know this triangle is isosceles.

prabsahi · Answer

Hi Siddharth,

Even I had this confusion .But, I see that holds true when you have an  Isosceles or Equilateral Triangle

So,basically if A line is drwan from Vertex of an equilateral triangle or an isosceles triangle to the opposite side such that it divides the base in 
in two equal sides (basically acts as median) then the angle will be perpendicular and equal as you mentioned.

From the facts in the question we dont know if its an isosceles or equalateral since we dont know two of its sides

Here is the text from the link you shared.I have highligted the one that states the same.

In an isosceles triangle  (where base is the side which is not equal to any other side):

- the altitude drawn to the base is the median and the angle bisector;

-  the median drawn to the base is the altitude and the angle bisector;
 
- the bisector of the angle opposite to the base is the altitude and the median.

Hope its helps!!

If AC=12, does ACB=90?

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If AC=12, does ACB=90?

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