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# What are the coordinates of point B in the xy-plane above ?

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Re: What are the coordinates of point B in the xy-plane above ? [#permalink]
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shinrai15 wrote:
Skywalker:

How did u get AC as 28?

Did u find the slope between AC

S

Hi shinrai,
I found AC using the coordinates of A (-8,0) and C (20,0).
You can use the distance formula but since Y coordinates are 0 , you can simply subtract.

Hope this helps!!
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Re: What are the coordinates of point B in the xy-plane above ? [#permalink]
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Simply using the information given to derive some facts can help a lot.

I'm a visual learner so I don't like reading a bunch of text explaining why things are the way they are.

So just quickly in relation to my diagram

We are told AB = BC
Therefore <ADB must be 90 degrees also

We are told AC = BD = 28 so write this in
This is common to both triangles, so <BAD = <BCA

Since two angles and two sides are equal the third angle (marked in green) and corresponding side must also be equal.

This allows us to conclude that D must be the midpoint of AC as the | must each equal each other

From here we can ascertain the midpoint as (20+(-8))/2 = 12/2 = 6

The y coordinate must be 28 since the height, BD, is 28

Thus, (6,28)
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Re: What are the coordinates of point B in the xy-plane above ? [#permalink]
Skywalker:

How did u get AC as 28?

Did u find the slope between AC

S
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Re: What are the coordinates of point B in the xy-plane above ? [#permalink]
Use distance formulae AB=BC given
Let point B be (x,y)
so, sqrt[(x-(-8))^2+(y-0)^2]=sqrt[(20-x)^2+(0-y)^2]
or, 16x+64=400-40x
or, x=6

This gives coordinate of D i.e. (6,0)
Given, AC=BD
use again distance formulae.
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Re: What are the coordinates of point B in the xy-plane above ? [#permalink]
question can be answered pretty quickly with visualization.

We are Given that AC = BD.

Length of AC = Distance along the X-Axis = [-8 - 20] = 28 Units

thus, Length of Altitude BD = 28 from the Vertex B to Opposite Side/Base AC

thus, the Y-Coordinate of Point B must be: Y = 28

Leaves only (B) and (E) as possible Answers

Rule: in an Isosceles Triangle, the Altitude dropped from the Vertex between the Equal Sides will act as the Line of Symmetry for the Isosceles Triangle. This Altitude will Bisect the NON-Equal Side.

From this Rule: AD = DC = (1/2) * 28 = 14 Units

The X-Coordinate of B will be the Length along the X-Axis from the Origin (0 , 0) to Point D, which forms a 90 Degree Angle with Height BD

Since we already know from above that AD = 14 Units, it would be Impossible for the Distance from the Origin (0 , 0) to Point D on the X-Axis to be 14 Units also. Therefore, it must be 6 Units, making the X-Coordinate of Point B (and Point D) = +6

Therefore, that leaves only Answer (B) 6,28
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Re: What are the coordinates of point B in the xy-plane above ? [#permalink]
Distance between A to C = 28.
Since AB=BC (isosceles triangle). The altitude is a perpendicular bisector.

So X intercept would be at 14 points away from -8 i.e. point 6 on x coordinate.

BD=28 points (Since AC=BD), hence altitude would be at 28 points.

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Re: What are the coordinates of point B in the xy-plane above ? [#permalink]
Bunuel wrote:

What are the coordinates of point B in the xy-plane above ?

(A) (6, 12)
(B) (6, 28)
(C) (8, 20)
(D) (12, 20)
(E) (14, 28)

Attachment:
Untitled-1.jpg

Solution:

Since AB = BC, triangle ABC is an isosceles triangle, and BD, the altitude, divides AC, the base, into two equal parts. In other words, AD = DC, and D is the midpoint of AC. Therefore, the x-coordinate of point D is (-8 + 20)/2 = 6, which is also the x-coordinate of point B (notice that we can eliminate choices C, D, and E). Lastly, since AC = BD, and the length of AC is 20 - (-8) = 28, the y-coordinate of point B is the y-coordinate of D plus 28. In other words, the y-coordinate of point B is 0 + 28 = 28. Therefore, the coordinates of point B are (6, 28).

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Re: What are the coordinates of point B in the xy-plane above ? [#permalink]
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Re: What are the coordinates of point B in the xy-plane above ? [#permalink]
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