In quadrilateral ABCD, AB = AD = 5 and BC = 6 and CD = 1. Which of the

Question

In quadrilateral ABCD, AB = AD = 5 and BC = 6 and CD = 1. Which of the following could be the value of angle A? (In degrees)

A 45
B 60
C 75
D 90
E 120

chetan2u · Accepted Answer

It all depends on the triangle that can be constructed by the two sides AB and AD, and the diagonal BD. But BD will get restricted by the two sides of triangle BCD. BD cannot be greater than sum of other two sides so <6+1 or <7, and cannot be lesser than the difference of other two sides, that is >6-1 or >5. Now let’s get back to the triangle ABD. 1) Least value of angle A is when we take least value of opposite side BD. So the triangle becomes 5,5,>5. Almost an equilateral triangle but with the side BD slightly more, so the opposite angle A will be slightly more than 60. 2) Maximum value of A. Say A is 90, then BD becomes hypotenuse and would be equal to \sqrt{5^2+5^2}=5\sqrt{2}=5*1.414=7 But BD <6, so A has to be less than 90. Range : 60

bb · Answer

Moving this question to the proper forum.

AndrewN · Answer

Hello, everyone. I have nothing to add to the excellent analysis provided above by chetan2u. However, for the more visually inclined (or dependent) solver, such as me, I thought I would attach a figure. When you pair the two posts—again, if you need more than a description—the problem should be easier to grasp. Even though it plays on the simple notion of finding a valid side length given two other sides of a triangle, since those triangles are hidden here and we still have to conceptualize what sorts of angles would be possible within one of them, I would probably call this a 700-level question. (Just my opinion.)

Attachment:

Screen Shot 2021-05-07 at 16.19.33.png [ 17.64 KiB | Viewed 9635 times ]

Cheers, and good luck with your studies.

- Andrew

Fdambro294 · Answer

The way I reasoned through it:

If we connect diagonal BD and call that side X, we can create triangle BCD inside the funky quadrilateral (sounds like a rap name from the 80s)

The triangle inequality theorem states that, when you have 1 unknown side:

(Differences between 2 known sides) < X < (SUM of two known sides)

The above inequality must hold for every triangle.

(6 - 1) < X < (6 + 1)

5 < X < 7

X is also the Non Equal side of an Isosceles Triangle ABD, with side lengths:

5
5
5 < X < 7

X can not be 5, so the Angle at the vertex between the 2 equal sides of length 5 can not be 60 degrees (this can’t be an equilateral triangle)

Also, if we assign any value from 5 to 7 (the boundaries) to Diagonal BD, the Pythagorean theorem can not be satisfied.

If Angle A is an Obtuse Angle greater than 90 degrees, then the following relationship will hold true (holds true for any obtuse triangle)

(Longest side)^2 > (5)^2 + (5)^2

Giving diagonal BD the highest boundary value of 7 (which it must be less than) we still can not satisfy this condition:

(7)^2 > (5)^2 + (5)^2
49 is not greater than 50

Therefore, triangle ABD can not be an obtuse triangle with an angle greater than 90 degrees. Each angle must be less than 90 degrees, including the angle at vertex A

Since BD is greater than 5, the angle at vertex A (opposite side BD) can not be 60 degrees. Side BD must be longer than 5, the length that would make it an equilateral triangle.

-if we start with the Angle at vertex A as 60 degrees, angle A will have to “stretch” wider to accommodate Side BD which is longer than 5

60 degrees < angle A < 90 degrees

Answer must be 75 degrees

Perfectly crafted question, answers and all.

Posted from my mobile device

bumpbot · Answer

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In quadrilateral ABCD, AB = AD = 5 and BC = 6 and CD = 1. Which of the

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GMAT Problem Solving (PS) Questions

In quadrilateral ABCD, AB = AD = 5 and BC = 6 and CD = 1. Which of the

Prep Toolkit

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

More than Quant/Verbal Abilities: Speed vs. Accuracy

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My Rewards