Maths Problem........help!!?

Question

Plz solve this problem (with steps).....

A equilateral triangle has two vertices at the points (3,4) and (-2,3), Find the third vertex .

Answer 1

The distance between the 2 vertices is √(1² + 5²) = √26. You need a point that same distance from both vertices (there will be 2 solutions). Point is (x,y), so (leaving distance squared),

(x-3)² + (y-4)² = 26 and
(x+2)² + (y-3)² = 26

x² - 6x + 9 + y² - 8y + 16 = x² + 4x + 4 + y² - 6y + 9
-6x - 8y + 25 = 4x - 6y + 13
-10x - 2y = -12
5x + y = 6

By setting the 2 expressions equal to each other, we lost the 26 distance² and have just found the perpendicular bisector (the hard way). Line through (3,4) and (-2,3) has slope 1/5, so perpendicular should have slope -5 and pass through midpoint (1/2, 7/2), which would make it 5x + y = 12/2 = 6.

Solving that for y in terms of x, y = 6 - 5x. Plugging that in to one of the distance equations:

(x-3)² + (y-4)² = 26
(x-3)² + (6-5x-4)² = 26
(x-3)² + (2-5x)² = 26
x² - 6x + 9 + 4 - 20x + 25x² = 26
26x² - 26x -13 = 0
2x² - 2x - 1 = 0
x = 2/4 ± √(4 + 8)/4
x = 1/2 ± √3 / 2

for x = (1+√3)/2, y = 6 - 5(1+√3)/2 = 7/2 - 5√3 /2
for x = (1-√3)/2, y = 6 - 5(1-√3)/2 = 7/2 + 5√3 / 2

Answer 2

note, an algebraic solution:

Process:

1. Plot your line segment (3, 4) (-2, 3)
2. Find the slope of this line segment
3. Find the mid-point of this line segment
4. The Altitude of the triangle with have a perpendicular slope to the base
5. Find the length of the base line segment (this will be the length of each side)
6. You have a right triangle using the altitude and half the base, with the third side being the hypotenuse
6a. You know the length of two sides (the hypothenuse and half the base)
7. Use the pythagorean theorem to find the length of the altitude.
8. Use the distance formula for the altitude & the equation of a line for the altitude as a system to find the third vertex

Answer 3

One method is to use the distance formula for two points on a Cartesian coordinate.

Since equilateral triangle implies all sides are of equal length or all interior angles are equal (60 degrees).

Note: There will also be two possible answers due to symmetry.

Let the third vertices's be (x,y), so now we can solve for (x,y) by recognizing that the distance of this unknown point to either of the given point is equal. Also, we can find the length of each side by computing the distance with the two given points.

You will get three equations, should be enough info to solve the unknowns.

Another method is to use your trigonometry knowledge (sine and cosine laws).

Third method is to use grid papers and graph this use a ruler to get the right point by measurement.

Hope this helps.
KT

Answer 4

Compute the vector that goes from (-2, 3) to (3, 4). Use cos and sin with the vectors, through an angle of 60 degrees. Don't forget to use the magnitude of the vector to determine the length of the side of the triangle.

Also, there are two answers to this problem.

Answer 5

This is hard to explain. You need to have grid paper, map out the 2 points, then find the 3rd point to make a triangle with equal sides. With that said, there are 2 answers that will be right. Love math yet???

Answer 6

let two complex nos be 3+4i and -2+3i.so diff of the complex nos =5+i rotate this vector through an angle +-pi/3 that is multiply the vector by (cospi/3+-isin pi/3) and you will get the vertex as either (5-sqrt3)/2,(5sqrt3+1)/2 or (5+sqrt3)/2,(1-5sqrt3)/2
p.s. i have given here the general rule. the process is correct.please make sure that the answer is all right.

Answer 7

Length of triangle side = sqr(26) - use pythagoras
angle line between line joining vertices and x-axis = arctan(1/5) = 11.3°.
So the other vertex is at:
x = -2 +sqr(26) cos(11.3+60) = -0.3660
y=3 +sqr(26)sin(11.3+60) = 7.8301