A(0,2), B(1,-0.5), C(2,-3) are collinear or not and how?
Answers
Answer:
Yes I can. Let’s do each way suggested and then a better way.
The slopes of AB and AC should agree:
A(0,2),B(1,−0.5),C(2,−3)
slope(AB)=−0.5−21−0=−2.5
slope(AC)=−3−22−0=−2.5✓
Those agree, so we have collinearity.
An alternative that’s essentially the same but avoids division is the point-point form of a line through (a,b) and (c,d):
(c−a)(y−b)=(d−b)(x−a)
We can consider that the relationship between three points on a line, in our case (a,b)=(0,2),(c,d)=(1,−0.5),(x,y)=(2,−3). Checking.
(c−a)(y−b)=(1−0)(−3−2)=−5
(d−b)(x−a)=(−0.5−2)(2−0)=−5✓
Now to the distance formula. For collinear points, two of the distances should add two the third. Looking at the points we check |AB|+|BC|=|AC|. This is in general a pain because of the square roots; let’s try it.
AB2=(1−0)2+(−0.5−2)2=1+6.25=7.25=29/4
AC2=(2−0)2+(−3−2)2=29
BC2=(2−1)2+(−3−−0.5)2=29/4
This is collinear because we have 29/4−−−−√+29/4−−−−√=29−−√✓.
I prefer to deal with the squared sides directly. If P,Q,R are the squared sides of a triangle, it’s a degenerate triangle, three collinear points, when
±P−−√±Q−−√=±R−−√
for some combination of pluses and minuses. Squaring,
P+Q±2PQ−−−√=R
(R−P−Q)2=4PQ
That’s called the Triple Quad Formula and it allows us to check the squared lengths directly for collinearity. The assignment of the squared sides to P,Q,R doesn’t matter; we can choose a way that makes the arithmetic easier, say R=P=29/4, Q=29.
(R−P−Q)2=(29/4−29/4−29)2=292
4PQ=4(29/4)(29)=292✓
These agree so we have collinearity. With the TQF we don’t have to work with any square roots or even figure out which is the longest side to check collinearity.
A bonus from the TQF is the difference of the two sides is sixteen times the area of the triangle when the points aren’t collinear.
16Δ2=4PQ−(R−P−Q)2
That’s called Archimedes’ Theorem. It’s often better than Heron’s Formula.
Step-by-step explanation:
If the area of triangle is zero then it can be collinear.If not then it is not collinear.First calculate the area of triangle by using the formula then you can conclude whether it is collinear or not.