Re: What are the coordinates of point B in the xy-plane above?
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19 May 2024, 04:56
The Ultimate GRE Cheat Sheet has a GREAT chapter on triangle properties and 2D geometry - you will find loads of line and shape properties and formulas (some are quite uncommon and advanced!) that will help you solve these questions very quickly.
You can very easily solve this question if you are familiar with triangle properties, coordinate geometry and line equations.
Let's start with finding the x coordinate:
Realise that AC is a horizontal line - both y coordinates are the same.
Realise that BD is the perpendicular bisector of triangle ABC and therefore that D is the midpoint of line AC- we are told that AB = BC and that D is perpendicular to B (denoted by the right angle shown in the diagram).
Because we know the length of AC, we can use the line midpoint formula to find the coordinates of D.
And, because we know B is perpendicular to D, we know that they will share the same x coordinate.
The formula to find coordinates of the midpoint of a line given the coordinates of two points:
((x1 + x2) / 2), ((y1 + y2) /2)
So the midpoint of AC is:
(-8+20)/2 = (12/2) = 6
,
(0 + 0)/2 = 0
gives us coordinates for point D (6,0). So, we know B also has x coordinate 6.
Now let's find the y coordinate:
We know that the base (AD) of triangle ABD is 14 units long (A is 8 units away from 0. D is 6 units away from 0. So, 8+6 = 14).
We know that AD is half the length of AC .
Therefore, AC = 2*14 = 28.
We are told that AC = BD.
Therefore we know that BD = 28 units long.
Because we know that the line AC lies on the x axis and that B is perpendicular to the x axis, we know that B is 28 units away from the x axis along the y axis.
Therefore, the y axis is 28.
So, the coordinates of B = (6, 28).
The answer is B.