What is the area of the triangle in the figure above?

If we sketch the altitude (i.e., height) from the 105-degree angle to the opposite side, we have created two special right triangles: a 0-60-90 right triangle at the top and a 45-45-90 right triangle at the bottom (and the only labeled side will be the hypotenuse of the 45-45-90 right triangle) .

Since the 45-45-90 right triangle at the bottom has hypotenuse 2√2, each of its legs is 2. Since one of these legs is the height of the overall triangle, it’s also the side opposite the 30-degree angle of the 30-60-90 right triangle at the top. So the side opposite the 60-degree angle is 2√3.

Now, using the fact that the area of a right triangle is ½ the product of its two legs, the area of the 45-45-90 right triangle is ½ x 2 x 2 = 2, and the area of 30-60-90 right triangle is ½ x 2 x 2√3 = 2√3. Therefore, the area of the overall triangle is 2 + 2√3.

Answer: A

Can someone suggest mistake in below solution.

Suppose Triangle is ABC with Angle A1 = 30, B = 105 and C = 45. Now, to calculate height, we draw one perpendicular line AD from A on BC (Extended). So now Triangle ADC is a right angle (45-90-45) and other triangle is ADB (15-90-75).
Now come to line DC which is having B as one point on it. Angle DAB = 15 and BAC = 30, So DB : BC = 15 : 30 = 1:2. BC is given = 2 * sq. root 2. So total DC = BC *3/2 = 3 * Sq. root 3.

As ADC is a 90-45-45 triangle, so AD will also be 3 * Sq. root3.

Now Triangle ABC. Base = BC = 2 * sq. root 2 and height = AD = 3 * Sq. root3

Area = (2 * sq. root 2) * (3 * Sq. root3) /2 = 6.


Can someone explain, what principle is wrong in above calculation.

Deepak

Here's a solution using the trigonometry table:

in question 4 why base 2 under root 2 is used rather than 2

Posted from my mobile device

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.