Two Sides of a Triangle and their Included Angles are 4cm, 5cm and 30 GMAT Problem Solving

Question: Two sides of a triangle and their included angles are 4cm, 5cm and 30 degree resp. What is the area of the triangle?

1. 10
2. 15
3. 5
4. 20

Correct Answer: C
Solution and Explanation:
Approach Solution 1:
The problem statement states that:
Given:

• Two sides of a triangle and their included angles are 4cm, 5cm and 30 degrees respectively.

Find out:

• The area of the triangle.

Let’s assume the side of the length 5 as the base of the triangle.
Imagine a height is drawn using this base of the triangle, then we will get a 30-60-90 triangle with a hypotenuse of 4.
Since the short side in this triangle is our height, and since the short side in a 30-60-90 triangle is half the hypotenuse, the height is 2.
As per the formula of a right-angled triangle, we know that area of the triangle = ½ * base * height.
Therefore, in this case, the area of this right-angled triangle = ½ * 5 * 2 = 5.
Hence, the area of the triangle = 5 sq. cm

Approach Solution 2:
The problem statement informs that:
Given:

• Two sides of a triangle and their included angles are 4cm, 5cm and 30 degrees respectively.

Find out:

• The area of the triangle.

We know that the area of a triangle with sides a and b and the included angle C is as follows:
Area = ½ ab sin C, where sin C means “the sine of angle C.”
Therefore, the area of the triangle in this problem is:
Area = ½ * 4 * 5 * sin 30 = ½ * 20 * ½ = 5
Of course, to use this formula it is required to know the basic concept of trigonometry (for example, sin 30 = ½).
Hence, the area of the triangle = 5 sq. cm

Approach Solution 3:
The problem statement suggests that:
Given:

• Two sides of a triangle and their included angles are 4cm, 5cm and 30 degrees respectively.

Find out:

• The area of the triangle.

If we consider 5cm as the base of the triangle, we can imagine a triangle using 4cm as 1 side and an angle of 30 degrees between them.
To find the height of the triangle, it is required to drop a perpendicular from the 4cm side on a 5cm base.
The height of the triangle can be calculated since we know the angle = 30 degrees.
Therefore, the height of the triangle = sin 30 * 4cm = 2
Hence, the area of the traingle = 1/2 * 5 * 2 = 5 sq.cm

“Two sides of a triangle and their included angles are 4cm, 5cm and 30”- is a topic of the GMAT Quantitative reasoning section of the GMAT exam. To solve the GMAT Problem Solving questions, the candidates must have a basic knowledge of mathematics and calculations. The candidates can analyse varieties of questions from the GMAT Quant practice papers that will help them to improve their mathematical knowledge.

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