Advertisements
Advertisements
Question
If (0, −3) and (0, 3) are the two vertices of an equilateral triangle, find the coordinates of its third vertex.
Advertisements
Solution
Let the given points be A(0, −3) and B(0, 3). Suppose the coordinates of the third vertex be C(x, y).
Now, ∆ABC is an equilateral triangle.
∴ AB = BC = CA
Squaring on both sides, we get \[36 = x^2 + \left( y - 3 \right)^2 = x^2 + \left( y + 3 \right)^2\]
\[x^2 + \left( y - 3 \right)^2 = x^2 + \left( y + 3 \right)^2 \]
\[ \Rightarrow y^2 - 6y + 9 = y^2 + 6y + 9\]
\[ \Rightarrow - 12y = 0\]
\[ \Rightarrow y = 0\]
Putting y = 0 in
\[x^2 + \left( 0 - 3 \right)^2 = 36\]
\[ \Rightarrow x^2 = 36 - 9 = 27\]
\[ \Rightarrow x = \pm \sqrt{27} = \pm 3\sqrt{3}\]
Thus, the coordinates of the third vertex are
APPEARS IN
RELATED QUESTIONS
If two opposite vertices of a square are (5, 4) and (1, −6), find the coordinates of its remaining two vertices.
Find the coordinates of the circumcentre of the triangle whose vertices are (3, 0), (-1, -6) and (4, -1). Also, find its circumradius.
Prove that the points (0, 0), (5, 5) and (-5, 5) are the vertices of a right isosceles triangle.
Prove that (4, 3), (6, 4) (5, 6) and (3, 5) are the angular points of a square.
Find the area of quadrilateral PQRS whose vertices are P(-5, -3), Q(-4,-6),R(2, -3) and S(1,2).
If the points A(4,3) and B( x,5) lie on the circle with center O(2,3 ) find the value of x .
The ordinate of any point on x-axis is
If x is a positive integer such that the distance between points P (x, 2) and Q (3, −6) is 10 units, then x =
If the centroid of the triangle formed by the points (a, b), (b, c) and (c, a) is at the origin, then a3 + b3 + c3 =
If the centroid of the triangle formed by (7, x) (y, −6) and (9, 10) is at (6, 3), then (x, y) =
Find the point on the y-axis which is equidistant from the points (S, - 2) and (- 3, 2).

In the above figure, seg PA, seg QB and RC are perpendicular to seg AC. From the information given in the figure, prove that: `1/x + 1/y = 1/z`
Find the point on the y-axis which is equidistant from the points (5, −2) and (−3, 2).
The point R divides the line segment AB, where A(−4, 0) and B(0, 6) such that AR=34AB.">AR = `3/4`AB. Find the coordinates of R.
If the sum of X-coordinates of the vertices of a triangle is 12 and the sum of Y-coordinates is 9, then the coordinates of centroid are ______.
Point (–3, 5) lies in the ______.
Point (0, –7) lies ______.
The point at which the two coordinate axes meet is called the ______.
If the vertices of a parallelogram PQRS taken in order are P(3, 4), Q(–2, 3) and R(–3, –2), then the coordinates of its fourth vertex S are ______.
The distance of the point (–4, 3) from y-axis is ______.
