in an equilateral triangle RST, R has coordinates (0.0)and T has coordinate of (2a,0).Find the coordinates of S in terms of a.
By symmetry, S must lie on the vertical line x=a.
Let the coordinates of S be (a,y).
Since the length of one side of the triangle is (2a-0)=2a, we can calculate the distance of (a,y) to (0,0) and equate to 2a accordingly.
Using Pythagoras theorem,
(2a)=sqrt(a²+y²)
Solving for y:
y²=(2a)²-a²=3a²
y=(√3)a
To find the coordinates of point S in terms of 'a' in an equilateral triangle RST, we can use the fact that in an equilateral triangle, all three sides are equal in length and all three angles are equal to 60 degrees.
Given that point R has coordinates (0, 0) and point T has coordinates (2a, 0), we can find the length of one side of the equilateral triangle using the distance formula.
The distance formula between two points (x1, y1) and (x2, y2) is given by:
d = √((x2 - x1)^2 + (y2 - y1)^2)
Applying the distance formula to points R(0, 0) and T(2a, 0):
d = √((2a - 0)^2 + (0 - 0)^2)
d = √(4a^2 + 0)
d = 2a
Since all three sides of an equilateral triangle are equal, the length of RS and ST is also 2a.
Now, to find the coordinates of point S, consider that point S lies vertically above point T. Since RS is a horizontal line segment, the x-coordinate of point S will be the same as the x-coordinate of point T, which is 2a. However, the y-coordinate of point S will be the vertical distance from point T to point S.
Since RS and ST form a right angle, we can use the Pythagorean theorem to find the height of the equilateral triangle (the distance from T to S). The height can be found using the formula:
height = (√3 * side length) / 2
Substituting the values:
height = (√3 * 2a) / 2
height = √3a
Therefore, the y-coordinate of point S is √3a. Thus, the coordinates of point S are (2a, √3a).
In an equilateral triangle RST, if R has coordinates (0,0) and T has coordinates (2a,0), we can find the coordinates of S by considering the properties of an equilateral triangle.
Since R and T are at the same height (y=0), the centroid of the triangle (represented by point G) will also be at the same height, and it will divide the base (RT) into two equal parts.
Let's find the x-coordinate of the centroid G:
The x-coordinate of G is the average of the x-coordinates of R and T.
So, x-coordinate of G = (0 + 2a) / 2 = a.
Now, let's find the y-coordinate of G:
The y-coordinate of G is the average of the y-coordinates of R and T, which is zero.
Therefore, the y-coordinate of G is also zero.
Since S is the third vertex of the equilateral triangle, its y-coordinate will be the same as that of G, which is zero.
Thus, the coordinates of S are (a, 0).