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01 Sep 2020, 20:00
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D
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
95%
(hard)
Question Stats:
33% (01:57) correct
67%
(01:36)
wrong
based on 294
sessions
History
Date
Time
Result
Not Attempted Yet
A rectangle is divided into four regions with the areas of 1, 3, 4, and X, as shown above. What is the value of X?
A. 1
B. 2
C. 3
D. 4
E. Cannot be determined.
|
M37-103
Attachment:
2020-08-24_18-46-02.png
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Tanushka77
Current Student
01 Sep 2020, 20:46
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Two triangles (with areas 1 and 4) are similar because of angles.
Ratio of their areas=4:1
Then Ratio of their sides=2:1
Then draw a height.
(x*h)/2=1
Xh=2
The area of the rectangle then would be 3h*2x=6xh=12
Subtracting other regions we get 12-(1+3+4)=4
The answer is D
Posted from my mobile device
Ratio of their areas=4:1
Then Ratio of their sides=2:1
Then draw a height.
(x*h)/2=1
Xh=2
The area of the rectangle then would be 3h*2x=6xh=12
Subtracting other regions we get 12-(1+3+4)=4
The answer is D
Posted from my mobile device
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IMG_20200902_064338.jpg [ 3.13 MiB | Viewed 8022 times ]
01 Sep 2020, 21:09
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Let us take the triangles with area 1 and area 4. The two triangles will be similar since, one pair of angles are vertically opposite and the other angles are alternate interior angles, since the sides of rectangle are paralles.
Ratio of areas of the two triangles is 4:1.
Ratio of sides of the triangles is 2:1.
Let the side of the smaller triangle be y and height be h. so area =1/2 * y *h = 1
=> y*h=2
Now the side of the triangle with area 4 is 2y and height is 2h. So one side of the rectangle will be 2y and the other side will be h + 2h= 3h
Area of rectangle =3h * 2y =6hy = 6* (2) (since y*h =2)
= 12
Area of rectangle is 12.
so area of x = 12 - ( 1 + 3 + 4)
= 12 - 8
=4
So, D is the correct answer.
Ratio of areas of the two triangles is 4:1.
Ratio of sides of the triangles is 2:1.
Let the side of the smaller triangle be y and height be h. so area =1/2 * y *h = 1
=> y*h=2
Now the side of the triangle with area 4 is 2y and height is 2h. So one side of the rectangle will be 2y and the other side will be h + 2h= 3h
Area of rectangle =3h * 2y =6hy = 6* (2) (since y*h =2)
= 12
Area of rectangle is 12.
so area of x = 12 - ( 1 + 3 + 4)
= 12 - 8
=4
So, D is the correct answer.
09 Dec 2021, 04:35
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Official Solution:
A rectangle is divided into four regions with the areas of 1, 3, 4, and X, as shown above. What is the value of X? (Figure not drawn to scale.)
A. \(1\)
B. \(2\)
C. \(3\)
D. \(4\)
E. Cannot be determined
Consider yellow and green triangles below:
In these triangles, red angles are equal because they are a pair of opposite angles formed by intersecting lines and blue angles are equal because they are internal angles formed by a line crossing two parallel lines (opposite sides of a rectangle are parallel). So, all three angles in these triangles are equal, which makes these triangles 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 area of green triangle to that of red triangle is \(4:1\), then the ratio of their corresponding sides must be \(2:1\). So, if the base of the yellow triangle is \(a\), then the base of the green triangle is \(2a\) and if the height of the yellow triangle is \(b\), then the height of the green triangle is \(2b\).
We are given that the area of the yellow triangle is 1, so \(\frac{1}{2}*ab=1\) and thus, \(ab=2\).
Next, notice that the length and width of the rectangle in this case would be \(2a\) and \(b+2b=3b\), and the area would be \((2a)*(3b)=6ab\). Since \(ab=2\), then the area of the rectangle is \(6ab=12\).
Finally, the unknown area is \(X=12-(4+3+1)=4\).
Answer: D
A rectangle is divided into four regions with the areas of 1, 3, 4, and X, as shown above. What is the value of X? (Figure not drawn to scale.)
A. \(1\)
B. \(2\)
C. \(3\)
D. \(4\)
E. Cannot be determined
Consider yellow and green triangles below:
In these triangles, red angles are equal because they are a pair of opposite angles formed by intersecting lines and blue angles are equal because they are internal angles formed by a line crossing two parallel lines (opposite sides of a rectangle are parallel). So, all three angles in these triangles are equal, which makes these triangles 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 area of green triangle to that of red triangle is \(4:1\), then the ratio of their corresponding sides must be \(2:1\). So, if the base of the yellow triangle is \(a\), then the base of the green triangle is \(2a\) and if the height of the yellow triangle is \(b\), then the height of the green triangle is \(2b\).
We are given that the area of the yellow triangle is 1, so \(\frac{1}{2}*ab=1\) and thus, \(ab=2\).
Next, notice that the length and width of the rectangle in this case would be \(2a\) and \(b+2b=3b\), and the area would be \((2a)*(3b)=6ab\). Since \(ab=2\), then the area of the rectangle is \(6ab=12\).
Finally, the unknown area is \(X=12-(4+3+1)=4\).
Answer: D
