In the figure, triangles ABC and ABD are right triangles. What is the

Question

https://gmatclub.com/forum/download/file.php?id=44362 In the figure, triangles ABC and ABD are right triangles. What is the value of x ? (A) 20 (B) 30 (C) 50 (D) 70 (E) 90 Source: Nova GMAT Difficulty Level: 550 triangle (1).jpg

GMATinsight · Answer

For ABC to be right angle, angle C must be 90º

i.e. Angle ABD also must be 90º for the given geometry to exist

i.e.  x = 90-20 = 70º

Answer: Option D

BrentGMATPrepNow · Answer

Let's add variables for a few more angles:
 https://i.imgur.com/VIIBp3q.png

GIVEN: ∆ABC is a right triangle. 
So, either ∠w or ∠v is 90°
However, if ∠w = 90° then ∆ABC CANNOT be a right triangle. 
Why not? 
Well, if ∠w = 90°, then ∠ABD must be GREATER THAN 90°, and since the 3 angles in ∆ABC must add to 180°, the other two angles (∠BAC and ∠BDC) must be less than 90°

So, we can be certain that ∠w does NOT equal 90°, which means ∠v = 90° 
If ∠v = 90° and we know that ∠BAC = 20°, then w = 70° (since all 3 angles add to 180°)

GIVEN: ∆ABD is a right triangle. 
If ∠v = 90°, then ∠u = 90°, which means ∠x cannot also equal 90° 
So, in order for ∆ABD to be a right triangle, ∠ABD must equal 90°
If ∠ABD = 90 and ∠BAD = 20°, then x = 70° (since all 3 angles add to 180°)

Answer: D

Cheers,
Brent

Pritishd · Answer

In my observation, problems of this nature are best and fastest solved by rotating the diagram to a orientation that we are best acquainted with. In this case the problem becomes very easy once we rotate the figure to orient the right angle at the bottom left. Have a look at the below diagram.

Also given in the prompt is that BCA is a right triangle.

https://hosting.photobucket.com/images/a246/Pritishd/0/81a781d9-bf44-407a-b208-431bea144977-original.jpg?width=1920&height=1080&fit=bounds

Ans.  D  i.e. 70

ScottTargetTestPrep · Answer

Solution:
 
We are given that triangle ABC is a right triangle. If the right angle of triangle ABC is at B (i.e., angle ABC = 90 degrees), then angle ABD of triangle ABD must be an obtuse angle since it’s greater than angle ABC. However, triangle ABD is also a right triangle, so it can’t have an obtuse angle. Therefore, the right angle of triangle ABC can’t be at B. In that case, it must be at C (i.e., angle ACB = 90 degrees), and therefore, angle ABC = 180 - 90 - 20 = 70 degrees. Since angle ABD is greater than 70 degrees, it could be the right angle of triangle ABD. If it’s not, then angle BDA is the right angle of triangle ABD. Notice that if the former is true, then x = 180 - 90 - 20 = 70 degrees, and if the latter is true, then x = 90 degrees. We will prove that only the former (angle ABD is a right angle) can be true. That is, it’s impossible for angle BDA to be a right angle.

Recall that we have established that angle ABD is greater than 70 degrees. Now, if angle BDA were a right angle, then even if angle ABD were just a little bit greater than 70 degrees, for example 71 degrees, then the sum of the measures of the 3 angles of triangle ABD would be 20 + 71 + 90 = 181 degrees, which is not possible since we all know the sum should be 180 degrees. Therefore, angle BDA can’t be a right angle. This leaves us with only angle BDA being the right angle and therefore, x = 70 (mentioned above).

Answer: D

In the figure, triangles ABC and ABD are right triangles. What is the

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

GMAT Problem Solving (PS) Questions

In the figure, triangles ABC and ABD are right triangles. What is the

Prep Toolkit

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