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Question
Show that the points (0, –1), (8, 3), (6, 7) and (– 2, 3) are vertices of a rectangle.
Solution
Let the points be P(0, –1), Q(8, 3), R(6, 7), S(–2, 3)
Distance between two points= `sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2`
∴ By distance formula,
d(P, Q) = `sqrt((8 - 0)^2 + [3 - (-1)]^2`
= `sqrt((8 - 0)^2 + (3 + 1)^2`
= `sqrt(8^2 + 4^2)`
= `sqrt(64 + 16)`
= `sqrt(80)` ......(i)
d(Q, R) = `sqrt((6 - 8)^2 + (7 - 3)^2`
= `sqrt((-2)^2 + (4)^2`
= `sqrt(4 + 16)`
= `sqrt(20)` ......(ii)
d(R, S) = `sqrt([(-2) - 6]^2 + (3 - 7)^2`
= `sqrt((-8)^2 + (-4)^2`
= `sqrt(64 + 16)`
=`sqrt(80)` ......(iii)
d(P, S) = `sqrt([(-2) - 0]^2 + [3 - (-1)^2]`
= `sqrt((-2)^2+ (3+ 1)^2`
= `sqrt((-2)^2 + 4^2`
= `sqrt(4 + 16)`
= `sqrt(20)` ......(iv)
In ▢PQRS,
∴ side PQ = side RS .......[From (i) and (iii)]
side QR = side PS ......[From (ii) and (iv)]
∴ ▢PQRS is a parallelogram ......[A quadrilateral is a parallelogram, if both the pairs of its opposite sides are congruent]
d(P, R) = `sqrt((6 - 0)^2 + [7 - (-1)]^2`
= `sqrt((6 - 0)^2 + (7 + 1)^2`
= `sqrt(6^2 + 8^2)`
= `sqrt(36 + 64)`
= `sqrt(100)`
= 10 ......(iv)
d(Q, S) = `sqrt([(-2) - 8]^2 + [3 - 3]^2`
= `sqrt((-10)^2 + (0)^2`
= `sqrt(100 + 0)`
= `sqrt(100)`
= 10 ......(vi)
In parallelogram PQRS,
PR = QS .......[From (v) and (vi)]
∴ ▢PQRS is a rectangle. .......[A parallelogram is a rectangle if its diagonals are equal]
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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?
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In a GPS, The lines that run east-west are known as lines of latitude, and the lines running north-south are known as lines of longitude. The latitude and the longitude of a place are its coordinates and the distance formula is used to find the distance between two places. The distance between two parallel lines is approximately 150 km. A family from Uttar Pradesh planned a round trip from Lucknow (L) to Puri (P) via Bhuj (B) and Nashik (N) as shown in the given figure below.
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Based on the above information answer the following questions using the coordinate geometry.
- Find the distance between Lucknow (L) to Bhuj (B).
- If Kota (K), internally divide the line segment joining Lucknow (L) to Bhuj (B) into 3 : 2 then find the coordinate of Kota (K).
- Name the type of triangle formed by the places Lucknow (L), Nashik (N) and Puri (P)
[OR]
Find a place (point) on the longitude (y-axis) which is equidistant from the points Lucknow (L) and Puri (P).
