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Question
A(2, 5), B(-2, 4) and C(-2, 6) are the vertices of a triangle ABC. Prove that ABC is an isosceles triangle.
Solution
AB = `sqrt ((2 + 2)^2 + (5 - 4)^2) = sqrt (16 + 1) = sqrt 17` units
BC = `sqrt ((-2 + 2)^2 + (4 - 6)^2) = sqrt (0 + 4)` = 2 units
AC = `sqrt ((2 + 2)^2 + (5 - 6)^2) = sqrt (16 + 1) = sqrt 17` units
It can be seen that AB = AC
Hence , the given coordinates are the vertices of an isosceles triangle.
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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.
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- Midfielders: As shown by players E, F and G.
- Fullbacks: As shown by players H, I and J.
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Using the picture of a hockey field below, answer the questions that follow:
If a player P needs to be at equal distances from A and G, such that A, P and G are in straight line, then position of P will be given by ______.
A circle has its centre at the origin and a point P(5, 0) lies on it. The point Q(6, 8) lies outside the circle.
If (– 4, 3) and (4, 3) are two vertices of an equilateral triangle, find the coordinates of the third vertex, given that the origin lies in the interior of the triangle.
