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

Question

https://gmatclub.com/forum/download/file.php?id=28394 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) Untitled-1.jpg

generis · Accepted Answer

Given (AB = BC): This triangle is isosceles.

The altitude, BD, of an isosceles triangle is a perpendicular bisector of the opposite side, which creates two congruent right triangles with equal bases, AD and DC.

Because the altitude is both perpendicular and a bisector, the x-coordinate of D also will be the x-coordinate of B.

To find the x-coordinate of D, and hence of vertex B, use midpoint formula:

\frac{x_1 + x_2}{2}=\frac{-8 + 20}{2}=\frac{12}{2}=   6

To find y-coordinate of vertex B, use given information that AC = BD. To find  distance  between two points that lie on the same line (y = 0), subtract x-coordinates of A and C*:
 
20 - (-8) = (20 + 8) = 28 = AC
AC = BD = 28. B's y-coordinate =  28

B = (6,28)

Answer B

*  You also can just "count": From -8 to 0 = distance of 8. From 0 to 20 = distance of 20. (8 + 20) = 28

Skywalker18 · Answer

Triangles ABD and CDB are congruent (RHS congruence)
∠ BDA= ∠ BDC
AB = BC
BD=BD (Common side)

Therefore , AD=DC 
D is the midpoint of AC
D is (6,0)

Since AC = BD
We know AC = 28
Therefore B is (6,28)

Answer B

shinrai15 · Answer

Skywalker:

How did u get AC as 28?

Did u find the slope between AC

Please clarify!

Thanks in Advance!
S

Skywalker18 · Answer

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!!

dcummins · Answer

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)

Rakesh2019 · Answer

Use distance formulae AB=BC given
Let point B be (x,y) 
so, sqrt =sqrt 
or, 16x+64=400-40x
or, x=6

This gives coordinate of D i.e. (6,0)
Given, AC=BD
use again distance formulae.

Fdambro294 · Answer

question can be answered pretty quickly with visualization.

We are Given that AC = BD.

Length of AC = Distance along the X-Axis =   = 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

kovid231 · Answer

Distance between A to C = 28.
Since AB=BC (isosceles triangle). The altitude is a perpendicular bisector.

Hence, AD=DC=14 points
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.

Hence answer should be (6,28)

ScottTargetTestPrep · Answer

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).

Answer: B

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

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