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Re: In the rectangular coordinate plane shown, what are the coo [#permalink]
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The official explanation

From the figure, since EC is perpendicular to the x-axis, C is a point vertically below point B. Hence, both have the same x-coordinate.
The line AC is horizontal and therefore its length equals the x-coordinate difference of A and C, which equals 6 – 0 = 6.
The line DC is horizontal and therefore its length equals the x-coordinate difference of D and C, which is 6 – 2 = 4.
The length of the vertical line BC equals the y-coordinate difference of B and C, which is 3 – 0 = 3.
The length of the vertical line EC equals the y-coordinate difference of E and C.

Now, in ∆ABC, ∠A = a°, ∠B = 90° – a°, and ∠C = 90°. The sides opposite angles A and B are in the ratio BC/AC = 3/6 = 1/2.
Similarly, in ∆DEC, ∠E = a°, ∠D = 90° – a°, and ∠C = 90° (So, ABC and DEC are similar triangles and their corresponding sides are proportional).
Hence, the sides opposite angles E and D are in the ratio DC/EC = 1/2. Hence, we have DC/EC = 1/2 EC = 2DC EC = 2 ⋅ 4 = 8

Hence, the y-coordinate of point E is 8, and the answer is (E).
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Re: In the rectangular coordinate plane shown, what are the coo [#permalink]
Lets establish a few things at first:
In triangle ABC,
Perpendicular = AC , Base = BC , Hypotenuse = AB

In triangle CED,
Perpendicular = CE , Base = CD , Hypotenuse = DE

Now in triangle ABC,
AC = [(6-0)^2 + (0-0)^2]^0.5 = [6^2 + 0]^0.5 = 6
BC = [(6-6)^2 + (3-0)^2]^0.5 = [0 + 3^2]^0.5 = 3
AB = [(6-0)^2 + (3-0)^2]^0.5 = [6^2 + 3^2]^0.5 = [36+9]^0.5 = 45^0.5

And in triangle CED we can find CD only:
CD = [(6-2)^2 + (0-0)^2]^0.5 = [4^2 + 0]^0.5 = 4
The X-coordinate of E will be same as C, i.e. 6. The Y-coordinate of E can be termed as y for now

We know that sides opposite the same angles (90 degree angle and the a degree angle) correspond too. Therefore, we can say that:
Perpendicular of triangle ABC / Base of triangle ABC = Perpendicular of triangle CED / Base of triangle CED
6 / 3 = CE / 4
CE = (6 / 3) * 4 = 2 * 4
CE = 8

Finding the y coordinate of E,
CE = [(6-6)^2 + (y-0)^2]^0.5
8 = [0 + y^2]^0.5
So, y = 8

Therefore the coordinates of Point E are (6,8). Option E) is the answer.
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Re: In the rectangular coordinate plane shown, what are the coo [#permalink]
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