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प्रश्न
Show that P(–2, 2), Q(2, 2) and R(2, 7) are vertices of a right angled triangle.
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उत्तर
Distance between two points = `sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2`
By distance formula,
PQ = `sqrt([2 - (-2)]^2 + (2 - 2)^2`
= `sqrt((2 + 2)^2 + (0)^2`
= `sqrt((4)^2`
= 4 ...(i)
QR = `sqrt((2 - 2)^2 + (7 - 2)^2`
= `sqrt((0)^2 + (5)^2`
= `sqrt((5)^2`
= 5 ...(ii)
PR = `sqrt([2 -(-2)]^2 + (7 - 2)^2`
= `sqrt((2 + 2)^2 + (5)^2`
= `sqrt((4)^2 + (5)^2`
= `sqrt(16 + 25)`
= `sqrt(41)`
Now, PR2 = `(sqrt(41))^2`
= 41 ...(iii)
Consider, PQ2 + QR2
= 42 + 52
= 16 + 25
= 41 ...[From (i) and (ii)]
∴ PR2 = PQ2 + QR2 ...[From (iii)]
∴ ∆PQR is a right angled triangle. ...[Converse of Pythagoras theorem]
∴ Points P, Q, and R are the vertices of a right angled 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.
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?
∆ABC with vertices A(–2, 0), B(2, 0) and C(0, 2) is similar to ∆DEF with vertices D(–4, 0), E(4, 0) and F(0, 4).
The point A(2, 7) lies on the perpendicular bisector of line segment joining the points P(6, 5) and Q(0, – 4).
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)`
