Find the location of the third point required to make an equilateral triangle.

Question

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Answer 1

#1 A (0,0) C(3,0) B (?,?)

This is an equilateral triangle where the bottom side is flat along the x axis, and we are looking for the missing third point, the peak.

For an equilateral triangle, all angles are 60 degrees and all sides are the same length.

Since the line AC is 3 units long, all sides of this triangle must be 3. If this triangle is going to be equal on all sides, then the peak must appear exactly midway between these 2 points.

A(0,0) and C(3,0) the middle of 3 and 0 is

3+0 / 2 = 3/2 or 1.5 and the middle of 0 and 0 is 0

The x coordinate of the missing point B has to be 3/2 or 1.5 to land exactly halfway between the other two points. We will use trig to find the missing y coordinate

Triangle from A to B to the halfway point of A and C

hypotenuse (line segment AB) = 3

smaller leg (A to the halfway point) = 1.5

find the missing vertical leg —> this will tell us how far above our flat line the y coordinate of B is. You can do Pythagorean theorem or cosine.

1.5^2 + b^2 = 3^2

2.25 + b^2 = 9

b^2 = 6.75

b = sqrt6.75 = 3sqrt(3) /2 = 2.60

The missing y coordinate is 2.60 units above 0 = 2.60

Missing point = (3/2, 3sqrt(3)/2) or (1.5, 2.60)

#2

A(4,5), C(4,-3)

We will apply the same concept except this time the flat side of our triangle is vertical at x=4. We can use these two points to figure out what the length of each side needs to be.

4 to 4 = no change

5 to -3 = 8 change

Each side has a length of 8 and each angle is still 60 degrees

First, we will use the same principal and find the midpoint of AC.

x: 4+4 / 2 = 4

y: 5 + -3 / 2 = 1

We know that the missing peak has a y coordinate of 1.

Make a right triangle to find the x coordinate of the missing point.

Triangle A to B to halfway point of segment AC:

A to B is the hypotenuse of 8 since each side is 8 units

A to the halfway point of AC is from 5 to 1 = 4 units

The missing leg will tell us how far beside our flat line the c coordinate of B is.

4^2 + b^2 = 8^2

16 + b^2 = 64

b^2 = 48

b= sqrt(48) = 4sqrt3 = 6.93

The missing x coordinate is 6.93 units beside our flat line at x = 4.

4 + 4sqrt(3) or 4 + 6.93 = 10.93

Missing point = (10.93, 1)

****NOTE****

In both these cases, I added the length we found at the end onto each old x or y coordinate to get the final answer, but you would make a perfectly acceptable triangle by subtracting as well.

#1 answers: (1.5, 2.60) OR (1.5, -2.60)

#2 answers: (10.93, 1) OR (-2.93,1)

Answer 2

(0,0), (3,0) have a midpoint of (1.5,0) the 3rd point of an equilateral trianlge will have the same x coordinate.

(1.5, y) there are two possible values for y, one positive the other negative.

(1.5, y) to (0,0) = a distance of 3

3^2= 1.5^2 + y^2

y^2 = 9-2.25 = 6.75

y = sqr6.75

the 3rd point of an equilateral trianle is (1.5, sqr6.75) or (1.5, -sqr6.75) = about (3/2, + 2.6)

for

(4,5), (4, -3) the midpoint is (4, 1) the 3rd point will have a y coordinate of 1

(x,1)

(x-4)2 = 8^2 - (5-1)^2 = 64 -16 = 48

x-4 = sqr48

x =4sqr6 + 4 = about 10.9 or -2.9

3rd point is either (-2.9, 1) or (10.9, 1)