enigma123 wrote:
In the figure, point D divides side BC of triangle ABC into segments BD and DC of lengths 1 and 2 units respectively. Given that ÐADC = 60º and ÐABD = 45º, what is the measure of angle x in degrees? (Note:
Figure is not drawn to scale.)
(A) 55
(B) 60
(C) 70
(D) 75
(E) 90
Any idea guys what will be the correct answer please? Also any idea how can I cut and paste the pictures in my post? Is it possible?
Complete solution for all the angles is in the image below:
x=45+30=75.
Notes:
Sides with one blue segment crossing them are equal and sides with two blues segments crossing them are equal too.
CO is perpendicular to AD -->
OD=1 (from 30-60-90 right triangle property as sides are in ratio \(1:\sqrt{3}:2\)) --> as OD=BD=1 then
ODB is an isosceles triangle.
<CDO and <BDO are supplementary to each other (supplementary angles are two angles that add up to 180°), so
<BDO=120 -->
<DAB=180-(120+45)=15.
As ODB is an isosceles triangle -->
<DOB=<DBO=30.
<OBA=45-30=15 --> AOB is an isosceles triangle, so
OA=OB. Also COB is an isosceles triangle, so
CO=OB -->
OA=CO=OB. So, AOC is an isosceles triangle -->
<CAO=<OCA=45 (as <COA=90) -->
x=45+30=75.
Answer: D.
P.S. You can attach image files directly to the post.
Attachment:
Triangle complete.PNG
I have two questions which came in when I was solving the problem. Tried reading through the other explanations but am still unclear.
1. When you drew the segment OB, how did you determine that it will split angle B (i.e. 45) into 30 and 15? Why could it not be any other split like 20 and 25 or 10 and 35 etc?
2. In triangle AOC we know that angle O is 90, but then how did you conclude that the other 2 angles are 45 each and not any other split?
Cheers. Wishing Luck to Every GMAT Aspirant!