Advertisements
Advertisements
Question
Prove that the points A(–5, 4); B(–1, –2) and C(5, 2) are the vertices of an isosceles right-angled triangle. Find the co-ordinates of D so that ABCD is a square.
Solution
We have:
`AB = sqrt((-1 + 5)^2 + (-2 - 4)^2)`
= `sqrt(16 + 36)`
= `sqrt(52)`
`BC = sqrt((-1 + 5)^2 + (-2 - 2)^2)`
= `sqrt(36 + 16)`
= `sqrt(52)`
`AC = sqrt((5 + 5)^2 + (2 - 4)^2)`
= `sqrt(100 + 4)`
= `sqrt(104)`
AB2 + BC2 = 52 + 52 = 104
AC2 = 104
∵ AB = BC and AB2 + BC2 = AC2
∵ ABC is an isosceles right-angled triangle.
Let the coordinates of D be (x, y).
If ABCD is a square, then,
Mid-point of AC = Mid-point of BD
`((-5 + 5)/(2),(4 + 2)/(2)) = ((x - 1)/(2),(y - 2)/(2))`
`0 = (x - 1)/2, 3 = (y - 2)/2`
x = 1, y = 8
Thus, the co-ordinates of point D are (1, 8).
APPEARS IN
RELATED QUESTIONS
Given M is the mid-point of AB, find the co-ordinates of B; if A = (3, –1) and M = (–1, 3).
Find the coordinates of the midpoint of the line segment joining P(0, 6) and Q(12, 20).
Find the midpoint of the line segment joining the following pair of point :
(4,7) and (10,15)
The points (2, -1), (-1, 4) and (-2, 2) are midpoints of the sides ofa triangle. Find its vertices.
Let A(-a, 0), B(0, a) and C(α , β) be the vertices of the L1 ABC and G be its centroid . Prove that
GA2 + GB2 + GC2 = `1/3` (AB2 + BC2 + CA2)
The coordinates of the end points of the diameter of a circle are (3, 1) and (7, 11). Find the coordinates of the centre of the circle.
A(3, 1), B(y, 4) and C(1, x) are vertices of a triangle ABC. P, Q and R are mid - points of sides BC, CA and AB respectively. Show that the centroid of ΔPQR is the same as the centroid ΔABC.
The points A(−3, 6), B(0, 7) and C(1, 9) are the mid-points of the sides DE, EF and FD of a triangle DEF. Show that the quadrilateral ABCD is a parallelogram.
If the coordinates of one end of a diameter of a circle is (3, 4) and the coordinates of its centre is (−3, 2), then the coordinate of the other end of the diameter is
ABC is a triangle whose vertices are A(1, –1), B(0, 4) and C(– 6, 4). D is the midpoint of BC. Find the coordinates of D.
