Advertisements
Advertisements
Question
Prove that the points P (0, -4), Q (6, 2), R (3, 5) and S (-3, -1) are the vertices of a rectangle PQRS.
Advertisements
Solution
PQ = `sqrt((6 - 0)^2 + (2 + 4)^2) = 6sqrt(2)"units"`
QR = `sqrt((6 -3)^2 + (2 - 5)^2) = 3sqrt(2)"units"`
RS = `sqrt((3 +3)^2 + (5 + 1)^2) = 6sqrt(2)"units"`
PS = `sqrt((-3 - 0)^2 + (-1 + 4)^2) = 3sqrt(2)"units"`
PR = `sqrt((3 - 0)^2 + (5 + 4)^2) = 3sqrt(10)"units"`
QS = `sqrt((6 +3)^2 + (2 + 1)^2) = 3sqrt(10)"units"`
∵ PQ = RS and QR = PS,
Also PR = QS
∴ PQRS is a rectangle.
APPEARS IN
RELATED QUESTIONS
Find the distance between the points (0, 0) and (36, 15). Can you now find the distance between the two towns A and B discussed in Section 7.2.
If the point A(x,2) is equidistant form the points B(8,-2) and C(2,-2) , find the value of x. Also, find the value of x . Also, find the length of AB.
Find the distance between the following pair of point.
T(–3, 6), R(9, –10)
Find the distances between the following point.
A(a, 0), B(0, a)
Prove that the points (6 , -1) , (5 , 8) and (1 , 3) are the vertices of an isosceles triangle.
Prove taht the points (-2 , 1) , (-1 , 4) and (0 , 3) are the vertices of a right - angled triangle.
Prove that the points (0 , 2) , (1 , 1) , (4 , 4) and (3 , 5) are the vertices of a rectangle.
Show that the points (2, 0), (–2, 0), and (0, 2) are the vertices of a triangle. Also, a state with the reason for the type of triangle.
Points A (-3, -2), B (-6, a), C (-3, -4) and D (0, -1) are the vertices of quadrilateral ABCD; find a if 'a' is negative and AB = CD.
Case Study -2
A hockey field is the playing surface for the game of hockey. Historically, the game was played on natural turf (grass) but nowadays it is predominantly played on an artificial turf.
It is rectangular in shape - 100 yards by 60 yards. Goals consist of two upright posts placed equidistant from the centre of the backline, joined at the top by a horizontal crossbar. The inner edges of the posts must be 3.66 metres (4 yards) apart, and the lower edge of the crossbar must be 2.14 metres (7 feet) above the ground.
Each team plays with 11 players on the field during the game including the goalie. Positions you might play include -
- Forward: As shown by players A, B, C and D.
- Midfielders: As shown by players E, F and G.
- Fullbacks: As shown by players H, I and J.
- Goalie: As shown by player K.
Using the picture of a hockey field below, answer the questions that follow:
What are the coordinates of the position of a player Q such that his distance from K is twice his distance from E and K, Q and E are collinear?
