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11 Jan 2022, 06:45
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In the xy-plane, the rectangle ABCD is placed such that AB is parallel to the x-axis and CB is the parallel to the y-axis. A right triangle, ABE not shown, is drawn inside the rectangle, such that side AB is the hypotenuse of the triangle, and point E is on side CD of the rectangle. Which of the following could be the x-coordinate of point E?
(A) 2
(B) 2.5
(C) 3
(D) 4
(E) 5.25
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11 Jan 2022, 07:13
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Slope of two sides of right angled triangle shall be m and -1/m.
From diagram, it is clear than co-ordinates of points B and D are (1,3) and (6,1) respectively.
Now, since point E lies on side CD, the Y co-ordinate of point E is 3. There E -> (x,3).
Option A: Slope of AE= (3-1)/(2-1)=2; Slope of BE= (3-1)/(2-6)=-1/2.
For other options the slopes of AE and BE are not in m * (-1/m) format. So, A is the answer.
From diagram, it is clear than co-ordinates of points B and D are (1,3) and (6,1) respectively.
Now, since point E lies on side CD, the Y co-ordinate of point E is 3. There E -> (x,3).
Option A: Slope of AE= (3-1)/(2-1)=2; Slope of BE= (3-1)/(2-6)=-1/2.
For other options the slopes of AE and BE are not in m * (-1/m) format. So, A is the answer.
11 Jan 2022, 09:52
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Bunuel
A and B are at the same height above X-axis. Hence, the y-coordinate of B will be 1
C and B are at the same distance from the Y-axis. Hence, the x-coordinate of B will be 6
Thus, coordinate of B = (6,1)
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1.jpg [ 13.44 KiB | Viewed 2092 times ]
C, D and E are at the same height above X-axis. Hence, the y-coordinate of each will be 3
Let the x-coordinate of E be p
Thus, coordinate of E = (p,3)
Since angle AEB is \(90^0\), product of the slopes of AE and EB will be -1
Slope of AE = (3-1)/(p-1) = 2/(p-1)
Slope of EB = (3-1)/(p-6) = 2/(p-6)
Thus, we have:
\([2/(p-1)] x [2/(p-6)] = -1\)
\(=> (p-1)(p-6) = -4\)
\(=> p^2 - 7p + 6 = -4\)
\(=> p^2 - 7p + 10 = 0\)
\(=> (p-2)(p-5) = 0\)
=> p = 2 or 5
Note: Alternatively, we could have just checked with the answer options and found which value of p satisfies the equation
Answer A
