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Question
The perimeter of a triangle with vertices (0, 4), (0, 0) and (3, 0) is ______.
Options
5 units
12 units
10 units
11 units
`7 + sqrt(5)` units
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Solution
The perimeter of a triangle with vertices (0, 4), (0, 0) and (3, 0) is 12 units.
Explanation:
The vertices of a triangle are (0, 4), (0, 0) and (3, 0).
Now, perimeter of ΔAOB = Sum of the length of all its sides
= Distance between (OA + OB + AB)
Distance between the points (x1, y1) and (x2, y2) is given by,
d = `sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2)`
To find:
Distance between A(0, 4) and O(0, 0) + Distance between O(0, 0) and B(3, 0) + Distance between A(0, 4) and B(3, 0)
= `sqrt((0 - 0)^2 + (0 - 4)^2) + sqrt((3 - 0)^2 + (0 - 0)^2) + sqrt((3 - 0)^2 + (0 - 4)^2)`
= `sqrt(0 + 16) + sqrt(9 + 0) + sqrt((3)^2 + (4)^2`
= `4 + 3 + sqrt(9 + 16)`
= `7 + sqrt(25)`
= 7 + 5
= 12
Therefore, the required perimeter of the triangle is 12.
RELATED QUESTIONS
If A(5, 2), B(2, −2) and C(−2, t) are the vertices of a right angled triangle with ∠B = 90°, then find the value of t.
Name the type of quadrilateral formed, if any, by the following points, and give reasons for your answer:
(- 1, - 2), (1, 0), (- 1, 2), (- 3, 0)
Find a relation between x and y such that the point (x, y) is equidistant from the point (3, 6) and (−3, 4).
If the points (2, 1) and (1, -2) are equidistant from the point (x, y), show that x + 3y = 0.
Show that the points A (1, −2), B (3, 6), C (5, 10) and D (3, 2) are the vertices of a parallelogram.
Find the centre of the circle passing through (6, -6), (3, -7) and (3, 3)
Find the co-ordinates of points of trisection of the line segment joining the point (6, –9) and the origin.
Find the distances between the following point.
P(–6, –3), Q(–1, 9)
Find the distance of the following point from the origin :
(8 , 15)
P and Q are two points lying on the x - axis and the y-axis respectively . Find the coordinates of P and Q if the difference between the abscissa of P and the ordinates of Q is 1 and PQ is 5 units.
Prove that the points (7 , 10) , (-2 , 5) and (3 , -4) are vertices of an isosceles right angled triangle.
Prove that the points (0,3) , (4,3) and `(2, 3+2sqrt 3)` are the vertices of an equilateral triangle.
Show that the points (a, a), (-a, -a) and `(-asqrt(3), asqrt(3))` are the vertices of an equilateral triangle.
Show that A(1, 2), (1, 6), C(1 + 2`sqrt(3)`, 4) are vertices of an equilateral triangle.
Using distance formula decide whether the points (4, 3), (5, 1) and (1, 9) are collinear or not.
The distance between the point P(1, 4) and Q(4, 0) is ______.
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 ______.
If (– 4, 3) and (4, 3) are two vertices of an equilateral triangle, find the coordinates of the third vertex, given that the origin lies in the interior of the triangle.
