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Question
AB and AC are the two chords of a circle whose radius is r. If p and q are
the distance of chord AB and CD, from the centre respectively and if
AB = 2AC then proove that 4q2 = p2 + 3r2.
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Solution
Let AC = a then AB = 2a
seg OM ⊥ chord AC, seg ON ⊥ chord AB.
AM =MC = `a/2` and AN = NB = a
In Δ OMA and Δ ONA, By theorem of Pythagoras,
`AO^2 = AM^2 + MO^2`
`AO^2 = AN^2+ q^2` ....... (1)
`AO^2 = AN^2 + NO^2`
`AO^2 = a^2 + p^2` ........ (2)
From equation (1) and (2)
`(a/2)^2 + q^2 = a^2 + p^2`
`a^2/4 + q^2 = a^2 + p^2`
`a^2 + 4q^2 = 4a^2 + 4p^2`
`4q^2 = 3a^2 + 4p^2`
`4q^2 = p^2 + 3(a^2 + p^2)`
`4q^2 = p^2 + 3r^2 .... ("In" Δ ONA, r^2 = a^2 + p^2)`
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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 x axis equidistant from I and E is ______.
Points A(4, 3), B(6, 4), C(5, –6) and D(–3, 5) are the vertices of a parallelogram.
