Is triangle ABC a right-angled triangle, if BD=EC and angle ACB =60?

Question

https://gmatclub.com/forum/download/file.php?id=58736 Is triangle ABC a right-angled riangle , if BD=DC and \angle ACB =60 ? (1) AB is parallel to DE. (2) D is the midpoint of AC. Chetan's questions Untitled01.png

Neha2404 · Answer

IMP-D

Statement-1

If DE Is parellel to DE
Then angle BAC = angel EDC
ANgle EDC Is =30
So BAC is -30
Angle ABC=180-30-60
 =90
Sufficient

Statement 2
D is mid point of AC
Means AD =AC= BD = 2
And triangle BDC
Angle -CBD=60
And in angle D is 90
According to right angle property when a perpendicular draw on hypotension it divide equally
Sufficient

Posted from my mobile device

TestPrepUnlimited · Answer

Typically GMAT would word the question as "If BD=DC and  \angle ACB =60 , is triangle ABC a right-angled triangle?"

From the graph we are given  	riangle DEC  is a right triangle with  \angle C = 60 .

Statement 1:

We have  \angle DEC = \angle ABC = 90  so triangle ABC a right triangle. Sufficient.

Statement 2:

BD = DC = AD, so BD is half of AC. From Thales theorem it follows that ABC is a right triangle. Sufficient.

Ans: D

chetan2u · Answer

We are given that BD=DC and  \angle ACB =60

(1) AB is parallel to DE.
AB || DE. Since DE makes a perpendicular on BC, AB will also make a perpendicular on BC.
Thus ABC is a right-angled triangle.

(2) D is the midpoint of AC.
We are not required to know related theorems in GMAT, so there has to be a simpler way.
Let us take  	riangle BDC  =>  BD=DC , so  \angle DBC = \angle DCB = 60 , so  \angle BDC = 180-60-60=60 
Let us now take  	riangle BDA  =>  \angle BDA = 180-\angle BDC =180-60=120 , BD=DA , so  \angle DBA = \angle DAB = \frac{60}{2}=30 , so  \angle BDA=30 
 \angle ABC = \angle ABD+\angle DBC=60+30=90 
Thus ABC is a right-angled triangle.

D

CEdward · Answer

chetan2u

Beautiful solution. What I didn't know was the following for statement 2:

BD=DC and BD=DA

This allowed you to determine that you are looking at two isosceles. Can you elaborate more on how you knew this? That's why I wasn't able to solve because I didn't see that...

CEdward · Answer

Why does it necessarily follow that BD = DC = AD?

Is triangle ABC a right-angled triangle, if BD=EC and angle ACB =60?

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards