Question: In a triangle ABC, shown above, three lines go across point O, such that the lines are parallel to the sides of triangle ABC. If the areas of the yellow, green and blue regions shown are 1, 4 and 9 respectively, what it the area of triangle ABC?
- 24
- 28
- 32
- 36
- 54
“Hot Competition 3 Sep/8AM: In A Triangle ABC, Shown Above, Three Lines”- is a topic that is covered under the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide Quantitative Review”. The students need to be highly efficient in mathematical calculations in order to solve GMAT Problem Solving questions. The calculative mathematical problems that fall under the GMAT Quant topic in the problem-solving part should be decoded only with sound qualitative skills.
Solution and Explanation:
Approach Solution 1:
The problem statement informs that:
Given:
- 3 lines go across point O in such a way that the lines are parallel to the sides of the triangle ABC.
- The ratio of the areas of the triangles is 1:4:9
It is required to find out the area of triangle ABC.
As per the Side-Side-Side theorem of the triangle, we know that:
Since every three sides in every three triangles are parallel to each other, all these triangles will be similar to each other.
In similar triangles, it can be derived as:
If the sides are in the ratio m/n, then the areas of the triangles will be in the ratio (m/n)^2.
The ratio of the areas of triangles given is 1 : 4 : 9
Therefore, the ratio of the corresponding sides of the triangles will be 1:2:3.
Let the base of the yellow triangle be x, and then the base of the green triangle will be 2x and the base of the blue triangle will be 3x.
As per the image, the two grey portions are parallelograms, then we can say that their opposite sides are equal.
Therefore, the base of triangle ABC is equal to the sum of the three bases = x + 3x + 2x = 6x.
The image also represents the blue triangle as similar to the triangle ABC.
Therefore, the ratio of their bases will be 6x : 3x = 2 : 1
Thus, the ratio of their areas will be 4 : 1
The area of the blue triangle is 9, and then the area of triangle ABC will be 4 * 9 which is equal to 36.
Hence, D is the correct answer.
Correct Answer: D
Approach Solution 2:
The problem statement discloses that:
Given:
- 3 lines are drawn across point O in a way that the lines are parallel to the sides of the triangle ABC.
- The ratio of the areas of the triangles is 1:4:9
Find out
- The area of triangle ABC.
Since all three sides of the three triangles are parallel to each other, all three triangles are similar.
The ratio of the sides of the three triangles is equal to the ratio of the area of triangles which is
1 : 4 :9
By simplifying we get, the ratio of the area of triangles = 1 : 2 : 3
Therefore, if the base of the yellow triangle is a,
Then the base of green triangles will be 2a
And the base of the blue triangle will be 3a
And if the height of the yellow triangle is b;
Then the height of the green triangle will be 2b
And the height of blue triangle will be 3b
Therefore, the base of the main triangle will be the sum of the three bases that is equal to 6a
The total height of the main triangle will be the sum of the heights of the three triangles that is equal to 6b
Therefore, Area = 1/2 * 6a * 6 b = 18 ab (as per the formula of the area of a triangle that is
area = ½ * base * height)
It is given that 1/2 a*b = 1
=> ab = 2;
Therefore 18 ab = 36
Hence, D is the correct answer.
Correct Answer: D
Approach Solution 3:
The problem statement discloses that:
Given:
- 3 lines are passing through point O in such a way that the lines are parallel to the sides of the triangle ABC.
- The ratio of the areas of the triangles is 1:4:9
Asked
- What is the area of triangle ABC?
All the triangles that are green, yellow and blue are similar since the lines passed through O are parallel to the base of triangle ABC.
Therefore, since all sides of the triangles are equal, all angles of the triangle are also equal.
Assuming a, b and c as the side of yellow, green and blue triangles then we can say,
a^2 : b^2 : c^2 = 1: 4: 9
Therefore, a : b : c = 1 : 2 : 3
We can infer that, a:b:c = 1x:2x:3x
Hence, the base of triangle ABC will be equal to the sum of a, b and c that is a+b+c = 6x
Applying the square of the ratio of the side is equal to the area of triangle
By comparing with the blue triangle whose side is 3x
= (6x/3x)^2 = (Area of triangle ABC / 9)
Therefore, area of triangle ABC is 36
Hence, D is the correct answer.
Correct Answer: D
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