Triangles ABC, ACD, and ABD are All Isosceles Triangles. Point E (not GMAT Problem Solving

Question: Triangles ABC, ACD, and ABD are all isosceles triangles. Point E (not shown) is the mid-point of\(\overline{BD}\). If the ratio of the length of\(\overline{CE}\)to the length of\(C\)is equal to\(\sqrt3:2\), then what is the measure, in degrees, of\(\angle CAD\)?

1. 10
2. 15
3. 30
4. 45
5. 60

Answer:
Approach Solution (1):
Triangle BCE is congruent to Triangle DCE (BC = CD, BE = DE (E is mid-point of side BD), CE is common)
Therefore in angle BEC = angle DEC = 90 degree
In triangle BCE side BE = 1, BC = 2 and CE =\(\sqrt3\)(Pythagoras Theorem)
Angle CBD = Angle CDB = 60
Angle BEC = Angle DEC = 90
Angle BCE = Angle DCE = 30
Therefore Angle ACB = Angle ACD = 150 degrees (Angle ACB + ACD + BCD = 360)
Therefore in Triangle ACD
Angle CAD = Angle CDA = 15 (Isosceles Triangle)
Correct option: B

Approach Solution (2):
Line \(\frac{CE}{BC}={\sqrt3\over2}\)
In triangle BCE
Cos O = Base/Hypotenuse
Cos O =\({\sqrt3\over2}\)
Now according to trigonometry Cos 30 =\({\sqrt3\over2}\)
So angle BCE = 30 deg
Now exterior angle property of the triangle
Angle BCE = Angle BAC + Angle ABC
Since Triangle ACB is a isosceles triangle
So Angle BAC = Angle ABC
Therefore Angle BAC = Angle ABC = 15 deg
Same case is followed for triangle CAD and angle\(\angle CAD\) = 15 deg
Correct option: B

Approach Solution (3):
1st: The extended line from point C to point E on side BD bisects the side BD
Thus, the line AE is a median of the isosceles triangle. However, we cannot be sure whichs sides are the two equal sides and which side is the non-equal base side
2nd: We are given that side CE : BC is the ratio of\(\sqrt{3}:2\)
Using the trigonometric ratios, the cosine of an angle (X) is determined by = opposite sides / hypotenuse of the right angled triangle containing the corresponding angles
In the case of the ratio:\(\sqrt{3}:2\)
The cosine of (30 degrees) is equal to the ratio of the opposite side / hypotenuse in a 30-60-90 right triangle
For this reason, angle ECB must be = 30 degrees
From this point, you can determine the other angles using similar trigonometric ratios. The angle at angle E is 90 degrees and angle EBC = 60 degrees
You can perform the same analysis on the other triangle created by line CE: triangle ECD is also a 30-60-90 degree right triangle
3rd at this point, we know that line AE drawn from vertex A is both:
The median of side DB
And the altitude drawn to side DB
Therefore, line AE must be the line of symmetry for the isosceles triangle, making the sides AD and AB the two equal sides of the isosceles triangle
However, we still do not know which sides are the two equal sides within the isosceles triangles ACB and ACD
At this point, you can observe that the exterior angle at point C creates a straight line angle with line CE
Since the angle ECB = 30 degrees
The angles at: Angle DCA and Angle BCA must be 150 degrees
Since the base angles opposite the two equal sides must have the same measure , the remaining 4 unknown angles must be opposite the equal sides in each isosceles triangle
At this point, you can pick one of the isosceles triangles and solve for the interior angles.
Within triangle DCA:
Angle DCA = 150 degrees
Angle DAC = X
Angle CDA = X
X + X + 150 = 180
X = 15 degrees = Angle DCA
Correct option: B

“Triangles ABC, ACD, and ABD are all isosceles triangles. Point E (not shown) is the mid-point of\(\overline{BD}\). If the ratio of the length of\(\overline{CE}\)to the length of\(C\)is equal to\(\sqrt{3}:2\), then what is the measure, in degrees, of\(\angle CAD\)?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

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