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Question
Prove that the points A (1, -3), B (-3, 0) and C (4, 1) are the vertices of an isosceles right-angled triangle. Find the area of the triangle.
Solution
AB =`sqrt((-3 -1)^2 + (0 +3)^2) = sqrt(16+9) = sqrt(25)` = 5
BC =`sqrt((4 + 3)^2 + (1 +0)^2)= sqrt(49+1)= sqrt(50) = 5sqrt(2)`
CA =`sqrt((1 -4)^2 + (-3 - 1)^2) = sqrt(9 + 16) = sqrt(25)` = 5
∵ AB = CA
A, B, C are the vertices of an isosceless triangle.
AB2 + CA2 = 25 + 25 = 50
BC2 = `(5sqrt(2))^2` = 50
∴ AB2 + CA2 = BC2
Hence, A, B, C are the vertices of a right-angled triangle.
Hence, ΔABC is an isosceles right-angled triangle.
Area of ΔABC = `(1)/(2) xx "AB" xx "CA"`
= `(1)/(2) xx 5 xx 5`
= 12.5 sq.units
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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.
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:
The point on y axis equidistant from B and C is ______.
Point P(0, 2) is the point of intersection of y-axis and perpendicular bisector of line segment joining the points A(–1, 1) and B(3, 3).
If (a, b) is the mid-point of the line segment joining the points A(10, –6) and B(k, 4) and a – 2b = 18, find the value of k and the distance AB.
Find distance between points P(– 5, – 7) and Q(0, 3).
By distance formula,
PQ = `sqrt(square + (y_2 - y_1)^2`
= `sqrt(square + square)`
= `sqrt(square + square)`
= `sqrt(square + square)`
= `sqrt(125)`
= `5sqrt(5)`
