7. Using distance formula, show that the points
(1, -1), (5, 2) and (9, 5) are collinear.
Answers
Answer:
Let A=(1,-1) , B=(5,2) ,C=(9,5)
If they are co-linear then AB+BC=AC
AB=root (5-1)^2 +( 2+1)^2
=root (4)^2+(3)^2
=root16+9
=root25
then AB=5 units
BC =root(9-5)^2 +(5-2) ^2
=root(4)^2 +(3)^2
=root16+9 = root 25
BC=5units
AC=root (9-1)^2 +( 5+1)^2
=root (8)^2 +(6)^2
=root 64+36=root 100
=10units
AB+BC=AC
Hence proved
That ABC is co-linear
Step-by-step explanation:
let us consider the points are
P(1, -1)
Q(5, 2)
R(9, 5)
here , if the points P, Q, R are collinear then PQ + QR = PR
we know,
Distance = √[(y2 - y1)² + (x2 - x1)² ]
PQ + QR = PR
√[(2-(-1))²+(5-1)²]+√[(5-2)²+(9-5)²]
=√[(5-(-1))²+(9-1)²]
√[(2+1)²+(5-1)²]+√[(3)²+(4)²]
=√[(6)² + (8)²]
√[(3)² + (4)²]+√[9+16]= √[36 + 64]
√[9+16] + √[25] = √[100]
√25 + √25 = √100
5 + 5 = 10
10 = 10
LHS = RHS
therefor given points (1, -1), (5, 2) and (9, 5) are collinear.
_____________________________
Additional methods :
method 1]
we know if the three points are collinear then the area of triangle form by these points is zero
we know the formula to find the area of triangle when the coordinates of vertices are given,
Are of triangle = |Ax(By−Cy)+ Bx(Cy−A y)+ Cx(Ay−By)| / 2
let the coordinates of vertices are,
P(1, -1) = (Px ,Py)
Q(5, 2) = (Qx, Qy)
R(9, 5) = (Rx, Ry)
here the vertices are P, Q, R
therefor,the formula will be
Area of triangle PQR
=|Px(Qy - Ry)+Qx (Ry - Py)
+Rx(Py-Qy)| / 2
=|1(2-5) + 5 (5 - (-1)) + 9 (-1 - 2)| / 2
=|1 (-3) + 5 (6) + 9 (-3)| / 2
=|-3 + 30 - 27 | / 2
= | -30 + 30 | / 2
= 0 / 2 = 0
Area of triangle is Zero, hence the given points (1, -1), (5, 2) and (9, 5)
are collinear
method 2 ]
here, the points are ,
P(1, -1)
Q(5, 2)
R(9, 5)
if the points P , Q, R are collinear then the slopes form by the lines PQ and QR are equal
slope = (y2 - y1) / (x2 - x1)
slope of PQ = (2 - (-1))/(5 - 1)
= 3 / 4
slope of QR = (5 - 2 ) / (9 - 5)
= 3 / 4
slope of PQ = slope of QR
hence the points (1, -1), (5, 2) and (9, 5) are collinear.