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प्रश्न
Show that the points (2, 0), (–2, 0), and (0, 2) are the vertices of a triangle. Also, a state with the reason for the type of triangle.
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उत्तर
A ≡ (2, 0) ≡ (x1, y1)
B ≡ (–2 , 0) ≡ (x2, y2)
C ≡ (0, 2) ≡ (x3, y3)
∴ ABC form a triangle.
AB = `sqrt ((x_2 - x_1)^2 + ("y"_2 - "y"_1)^2)`
= `sqrt ((-2-2)^2 + (0 - 0)^2)`
= `sqrt ((-4)^2 + 0)`
= `sqrt(16)`
= 4 units
AC = `sqrt ((x_3 - x_1)^2 + ("y"_3 - "y"_1)^2)`
= `sqrt ((0-2)^2 + (2 - 0)^2)`
= `sqrt ((-2)^2 + (2)^2)`
= `sqrt (4 + 4)`
= `sqrt(8)`
= `2sqrt(2)` units
BC = `sqrt ((x_3 - x_2)^2 + ("y"_3 - "y"_2)^2)`
= `sqrt ((0-(-2)^2) + (2 - 0)^2)`
= `sqrt ((0 + 2)^2 + (2 - 0)^2)`
= `sqrt ((2)^2 + (2)^2)`
= `sqrt(8)`
= `2sqrt(2)` units
So, if side AC and side BC are equal then the triangle is an isosceles triangle.
AB2 = BC2 + AC2
`(4)^2 = (2sqrt2)^2 + (2sqrt(2))^2`
16 = 8 + 8
16 = 16
So, it is a right-angle isosceles triangle.
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Case Study -2
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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 ______.
The point P(–2, 4) lies on a circle of radius 6 and centre C(3, 5).
