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09 Dec 2021, 05:44
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In a triangle ABC, three lines intersect at point O, each parallel to one side of the triangle. If the yellow, green, and blue regions have areas of 1, 4, and 9, respectively, what is the area of triangle ABC? (Note: The figure is not drawn to scale.)
A. \(24\)
B. \(28\)
C. \(32\)
D. \(36\)
E. \(54\)
09 Dec 2021, 05:44
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Official Solution:
In a triangle ABC, three lines intersect at point O, each parallel to one side of the triangle. If the yellow, green, and blue regions have areas of 1, 4, and 9, respectively, what is the area of triangle ABC? (Note: The figure is not drawn to scale.)
A. \(24\)
B. \(28\)
C. \(32\)
D. \(36\)
E. \(54\)
Since all three sides in each of the three triangles are parallel to one another, these triangles are similar to each other.
In similar triangles, if the sides are in the ratio \(\frac{m}{n}\), the areas of the triangles are in the ratio \((\frac{m}{n})^2\). Since the ratio of the areas of the triangles is 1:4:9, the ratio of their corresponding sides must be \(1:2:3\). Therefore, if the base of the yellow triangle is \(x\), the base of the green triangle is 2x, and the base of the blue triangle is 3x.
Next, the two gray figures in the image above are parallelograms, so their opposite sides are equal, making the base of triangle ABC equal to \(x + 3x + 2x = 6x\).
Triangle ABC is also similar to the blue triangle, and since the ratio of their bases is \(6x:3x = 2:1\), the ratio of their areas must be \(4:1\).
Lastly, as the area of the blue triangle is 9, the area of triangle ABC is \(4 * 9 = 36\).
Answer: D
In a triangle ABC, three lines intersect at point O, each parallel to one side of the triangle. If the yellow, green, and blue regions have areas of 1, 4, and 9, respectively, what is the area of triangle ABC? (Note: The figure is not drawn to scale.)
A. \(24\)
B. \(28\)
C. \(32\)
D. \(36\)
E. \(54\)
Since all three sides in each of the three triangles are parallel to one another, these triangles are similar to each other.
In similar triangles, if the sides are in the ratio \(\frac{m}{n}\), the areas of the triangles are in the ratio \((\frac{m}{n})^2\). Since the ratio of the areas of the triangles is 1:4:9, the ratio of their corresponding sides must be \(1:2:3\). Therefore, if the base of the yellow triangle is \(x\), the base of the green triangle is 2x, and the base of the blue triangle is 3x.
Next, the two gray figures in the image above are parallelograms, so their opposite sides are equal, making the base of triangle ABC equal to \(x + 3x + 2x = 6x\).
Triangle ABC is also similar to the blue triangle, and since the ratio of their bases is \(6x:3x = 2:1\), the ratio of their areas must be \(4:1\).
Lastly, as the area of the blue triangle is 9, the area of triangle ABC is \(4 * 9 = 36\).
Answer: D
19 Apr 2023, 20:47
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Are we saying that the sides of the blue triangle is similar to that of ABC on the same grounds as we did before? i.e - They have parallel sides.
