One day, three friends Aakash, Aditya and Manoj went to Children Park to play some games.
While playing at one moment Aakash is at the point P, Aditya is at point A and Manoj is at point at B such that Aakash’s distance from Aditya and Manoj are equal. If the position of Aditya and Manoj are given as (1, 2) and (3, 8) respectively and area of
triangle PAB = 10 sq. m,
then find the coordinates of P. Also find the distance of Aakash from Aditya and Manoj.
Answers
Answer:
Here is the answer...
1) First we use the dist formula
2) It is given that the area of triangle is 10m sq ..so use the formula for area of Traingle..
The coordinates of P are ( 5 , 4) or ( -1 , 6) and Aakash Distance from Aditya & Manoj = √20 m
Step-by-step explanation:
Aditya point A ( 1 , 2)
Manoj point B ( 3 , 8)
Aakash Distance from Aditya & Manoj are equal
Let say coordinate of P = (x , y)
=> (x - 1)² + (y -2)² = (x - 3)² + (y - 8)²
=> x² - 2x + 1 + y² -4y + 4 = x² - 6x + 9 + y² - 16y + 64
=> 4x + 12y = 68
=> x + 3y = 17
Area of Triangle PAB P(x , y) A ( 1 , 2) , B ( 3 , 8)
= (1/2)| x(2 - 8) + 1 (8 - y) + 3(y - 2)|
= (1/2)| -6x + 8 - y + 3y - 6|
= (1/2) | -6x + 2y + 2|
= -3x + y + 1 or 3x - y - 1
-3x + y + 1 = 10
=> -3x + y = 9
x + 3y = 17
=> y = 6 , x = - 1
3x - y - 1 = 10
=> 3x- y = 11
x + 3y = 17
=> x = 5 & y = 4
Aakash Distance from Aditya & Manoj = √ (5 - 1)² + (4-2)² = √(5 - 3)² + (4 - 8)² = √20
or √ (-1 - 1)² + (6-2)² = √(-1 - 3)² + (6 - 8)² = √20
the coordinates of P are ( 5 , 4) or ( -1 , 6)
Aakash Distance from Aditya & Manoj = √20 m
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