The sides of a triangle are given by the equations 3x+4y=10 , 4x-3y=5 , and 7x+y+10=0 ; show that the origin lies within the triangle .
Answers
Origin lies with in triangle if The sides of a triangle are given by the equations 3x+4y=10 , 4x-3y=5 , and 7x+y+10=0
Step-by-step explanation:
3x+4y=10 Eq1
4x-3y=5 Eq2
7x+y+10=0 Eq3
Lets first find vertex of triangle formed by given lines
3*eq1 + 4 * eq2
=> 25x = 50
=> x = 2 , y = 1
(2 , 1)
Eq1 - 4*eq3
=> -25x - 40 = 10
=> x = - 2 , y = 4
(-2 , 4)
Eq2 + 3 * eq3
=> 25x + 30 = 5
=> x = - 1 , y = -3
(-1 , -3)
Area of Δ (2 , 1) , (-2 , 4) (-1 , -3)
= (1/2) | 2 ( 4 + 3) -2(-3 - 1) - 1(1 - 4)|
= (1/2) | 14 + 8 + 3|
= 25/2
if Origin (0 , 0) lies with in triangle
Then Sum of Area of Triangle of (2 , 1) , (-2 , 4) , (0 , 0) & (-1 , -3) (2 , 1) , (0, 0) & (-2 , 4) (-1 , -3) , (0 , 0) = 25/2
(1/2)| 2(4 -0) - 2(0 - 1) + 0(1 - 4) | + (1/2) | -1(1 - 0) + 2(0 + 3) + 0(-3 - 1)| + (1/2) | -2(-3 - 0) - 1(0 - 4) + 0(4 + 3)|
= (1/2)| 8 + 2 + 0 | + (1/2) | -1 + 6 + 0| + (1/2) | 6 + 4 + 0|
= (1/2) | 10 |+ (1/2) | 5| + (1/2) | 10 |
= 25/2
Hence Origin lies with in triangle
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