Rectangle ABCD has AB = 6 and BC = 3. Point M is chosen on side AB so

Question

Rectangle ABCD has AB = 6 and BC = 3. Point M is chosen on side AB so that angle AMD = angle CMD. What is the degree measure of angle AMD?

A. 15
B. 30
C. 45
D. 60
E. 75

Sanjeetgujrall · Answer

Can someone tell how to solve this?

I got 60 which is I guess is the trap answer

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m1033512 · Answer

IMO E ,

in the attached picture ,

in triangle DMC,

DC = MC =6 (sidea opposite to equal angles )

now in right angle triangle BMC ,

here I applied cos formula from trigonometry

cos $ = BC/MC =3/6 = 1/2 = cos $ = cos 60

$ = 60 
so angle BMC = 30

now 
angle AMD + DMC + 30 = 180

calculate AMD = 75

award kudos if helpful

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m1033512 · Answer

Sory guys , it is not giving me option to upload the picture which i drew

you can draw your self and assume angle MCB = $ and follow the above mentioned approach Posted from my mobile device

Sanjeetgujrall · Answer

Gmat doesnt expect us to use trigonometry....there must be another way to solve this

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cshreyasee · Answer

I believe it will be easier to solve this by the method of elimination - 
Remember that angle AMD + angle CMD + angle BMC = 180 degrees.
For option A:
angle BMC = 180 degrees - 2(15 degrees) = 150 degrees which is not possible.

For option B:
angle BMC = 180 degrees - 2(30 degrees) = 120 degrees which is not possible.

For option C:
angle BMC = 180 degrees - 2(45 degrees) = 90 degrees which is only possible if M and B coincide (hence discard)

For option D:
angle BMC = 180 degrees - 2(60 degrees) = 60 degrees.
In that case, triangle BMC is a 30-60-90 triangle (angle MBC = 90 degrees and angle BCM = 30 degrees)

This implies that triangle DMC is an equilateral triangle (each side is of length 6 cms).

Back to triangle BMC, the side opposite to angle MBC is 6 cms (=side MC) while the side opposite angle BMC is 3 cms (=side BC of the rectangle). This is incompatible with the properties of a 30-60-90 triangle.

Hence option E must be correct.

Bogei · Answer

First time posting - please excuse the format.

Let ∠AMC = ∠CMD = x,

From the triangle AMC, we know ∠ACM + ∠AMC + ∠MAC = 180, i.e. ∠ACM + x + 90 = 180 => ∠ACM = 90 - x

Since ABCD is a rectangle, we know that ∠ACM + ∠MCD is 90, which means that 90 - x + ∠MCD = 90 based on above. 
This would mean ∠ MCD = 90- (90-x) = x

Now observe the triangle MCD, we already know ∠CMD is x from the question, we derived that ∠MCD is also x, which means triangle MCD is an isosceles triangle.
We now know CD = MD = 6. Also observe MD is the hypotenuse of triangle MBD.

Now consider the right-angled triangle MBD: MD = 6, BD = 3. This triangle follows the 1:\sqrt{3}:2 ratio (30:60:90). This would mean ∠BMD is 30.

From there we set up the formula ∠AMC + ∠CMD + ∠BMD = 180 => x+x+30= 180. => x = 75. Answer E.

Hope this helps.

Bogei

Fdambro294 · Answer

Perfect analysis!

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laducminh199 · Answer

Posted from my mobile device   
Why angle AMC = angle CMD ??? They just give us that
 angle AMD= angle CMD
 can u explain

Thelegend2631 · Answer

This does not require any Calculations,
If you were to just draw the figure , you will notice angle ∠BMC be Smaller than 90 Degrees.

Two possibilities are 30,60 ( 180 - 75*2 , 180 - 60*2 )

If you were to assume ∠BMC = 60
THEN BM = AM = sqrt{3}

However, AM = 6 Hence ∠BMC = 60 is rules out.

Hence ∠BMC = 30 
angle AMD + angle CMD = 180 - 30 
angle AMD = angle CMD = 150/2 = 75

sarthakyadav18 · Answer

This is my first time posting a reply. Please let me know if this is correct.

We know that angle AMD = angle DMC. Let's take them as = x.
angle AMD = angle MDC = x, as they are vertically opposite angles between parallel lines.
Therefore, DC = MC = 6.

In triangle BMC, by pythagoras theorem, 
6^2 = 3^2 + BM^2
Thus, BM = 3*(3^(1/2))
By the property of right triangles, sides are in proportion of a 30:60:90 triangle. Thus angle BMC = 30.

So x + x + 30 = 180
Thus x = 75 i.e. E

wasario · Answer

Attaching some images, self-reference solution that anyone can feel free to use.

The first image is the original question in illustrated form.

https://i.imgur.com/nUS0hJKh.jpg

Based on this first image, we can come up with the second image which lets us fill in a lot of useful angle details. First, we get  ∠CMB = 180 - 2x  because angles in a line must add up to 180. Then because angles in a triangle add up to 180, we can calculate that  ∠MCB = 180 - 90 - (180 - 2x) = 180 - 90 - 180 + 2x = 2x - 90 . Then because we know that all corners of a rectangle are 90 degrees, we can calculate that  ∠MCD = 90 - (2x - 90) = 90 - 2x + 90 = 180 - 2x . Once again using the property of 180 degrees in a triangle, we can finally calculate that  ∠MDC = 180 - x - (180 - 2x) = 180 - x - 180 + 2x = x .

https://i.imgur.com/MG3RMcJh.jpg

This gives us useful information about the sides that we can see in the third image. Now that we know that triangle DMC has 2 of the same angles, it must be an isosceles triangle, and the sides opposite the equal angles are also equal in length. Since CD is 6 because it is a side of the rectangle, MC is also 6. Then, we can calculate that  MB = \sqrt{6^2 - 3^3} = 3\sqrt{3}  using the Pythagorean Theorem.

https://i.imgur.com/Bvo4fCLh.jpg

This triangle represents the properties of a 30-60-90 triangle, where the hypotenuse is  2s  (aka  s = 3 ), the longer non-hypotenuse side across from the 60 degree angle is  s\sqrt{3}  (aka  3\sqrt{3} )and the shorter side opposite the 30 degree angle is s (aka  3 ). Therefore, we know that ∠MCB is 60 degrees and ∠CMB is 30 degrees.

Finally, we can calculate for x. (you can use either ∠MCB or ∠CMB and it will produce the same result).

If using ∠CMB:
 180 - 2x = 30 
->  2x = 150 
->  x = 75

If using ∠MCB:
 2x - 90 = 60 
->  2x = 150 
->  x = 75

Answer choice E is correct. Hope this helps like it helped me!

samarpan.g28 · Answer

Here is how I did it.

Rectangle ABCD has AB = 6 and BC = 3. Point M is chosen on side AB so

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Rectangle ABCD has AB = 6 and BC = 3. Point M is chosen on side AB so

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