Question 1: Find the values of so that the point
lies inside or on the triangle formed by the lines
and
.
Answer:
Let the triangle be
. Let,
… … … i)
… … … ii)
… … … iii)
Solving i), ii) and iii) we get the vertices:
Given point
The point will lie inside or on
is the following three conditions hold hold simultaneously.
i) and
lie on the same side of
ii) and
lie on the same side of
iii) and
lie on the same side of
If and
lie on the same side of
then
… … … … … iv)
If and
lie on the same side of
then
… … … … … v)
If and
lie on the same side of
then
… … … … … vi)
From iv), v) and vi) we get
Question 2: Find the values of the parameter a so that the point is an-interior point of the triangle formed by the lines
and
.
Answer:
Let the triangle be
. Let,
… … … i)
… … … ii)
… … … iii)
Solving i), ii) and iii) we get the vertices:
Given point
The point will lie inside or on
is the following three conditions hold hold simultaneously.
i) and
lie on the same side of
ii) and
lie on the same side of
iii) and
lie on the same side of
If and
lie on the same side of
then
… … … … … iv)
If and
lie on the same side of
then
… … … … … v)
If and
lie on the same side of
then
… … … … … vi)
From iv), v) and vi) we get
Question 3: Determine whether the point lies inside or outside the triangle whose sides are given by the equations
.
Answer:
Let the triangle be
. Let,
… … … i)
… … … ii)
… … … iii)
Solving i), ii) and iii) we get the vertices:
Given point
The point will lie inside or on
is the following three conditions hold hold simultaneously.
i) and
lie on the same side of
ii) and
lie on the same side of
iii) and
lie on the same side of
If and
lie on the same side of
then
: This is FALSE
If and
lie on the same side of
then
: This is TRUE
If and
lie on the same side of
then
: This is FALSE
Therefore we can say that the point lies outside the
.