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प्रश्न
Name the type of triangle formed by the points A(–5, 6), B(–4, –2) and C(7, 5).
उत्तर
To find the type of triangle, first we determine the length of all three sides and see whatever condition of triangle is satisfy by these sides.
Now, using distance formula between two points,
AB = `sqrt((-4 + 5)^2 + (-2 - 6)^2` ...`[∵ d = sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2)]`
= `sqrt((1)^2 + (-8)^2`
= `sqrt(1 + 64)`
= `sqrt(65)`
BC = `sqrt((7 + 4)^2 + (5 + 2)^2`
= `sqrt((11)^2 + (7)^2`
= `sqrt(121 + 49)`
= `sqrt(170)`
And CA = `sqrt((-5 - 7)^2 + (6 - 5)^2`
= `sqrt((-12)^2 + (1)^2`
= `sqrt(144 + 1)`
= `sqrt(145)`
We see that,
AB ≠ BC ≠ CA
And not hold the condition of Pythagoras in a ΔABC.
i.e., (Hypotenuse)2 = (Base)2 + (Perpendicular)2
Hence, the required triangle is scalene because all of its sides are not equal i.e., different to each other.
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संबंधित प्रश्न
If A(4, 3), B(-1, y) and C(3, 4) are the vertices of a right triangle ABC, right-angled at A, then find the value of y.
Using the distance formula, show that the given points are collinear:
(6, 9), (0, 1) and (-6, -7)
ABC is an equilateral triangle . If the coordinates of A and B are (1 , 1) and (- 1 , -1) , find the coordinates of C.
What point on the x-axis is equidistant from the points (7, 6) and (-3, 4)?
Show that the point (11, – 2) is equidistant from (4, – 3) and (6, 3)
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:
If a player P needs to be at equal distances from A and G, such that A, P and G are in straight line, then position of P will be given by ______.
The distance of the point P(–6, 8) from the origin is ______.
A circle has its centre at the origin and a point P(5, 0) lies on it. The point Q(6, 8) lies outside the circle.
The point A(2, 7) lies on the perpendicular bisector of line segment joining the points P(6, 5) and Q(0, – 4).
The centre of a circle is (2a, a – 7). Find the values of a if the circle passes through the point (11, – 9) and has diameter `10sqrt(2)` units.
