Question

A triangle is formed by coordinates (0,0), (0,21) and (3,0). Find the number of integral coordinates that strictly lie inside the triangle(integral co-ordinates has both x and y as integers):
(a)19
(b)20
(c)21
(d)38

Answer 1

Hint: In this question, we have to draw the diagram so that we get a clear idea of what we have to find. Then find the line equations of the sides of the triangle and then check those coordinates that lie strictly inside the triangle.

Complete step-by-step answer:

Let us look at definitions first.

TRIANGLE: A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry.

STRAIGHT LINE: A straight line is the set of all those points which are collinear with the given points.

General equation of a line is \[ax+by+c=0\]

Two Points Form: The equation of a line passing through the points (x1,y1) and (x2,y2) is given by

\[\left( y-{{y}_{1}} \right)=\left( \dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}} \right)\left( x-{{x}_{1}} \right)...........\left( 1 \right)\]

Now , let us draw the diagram according to the given conditions (0,0), (0,21) and (3,0)

The equation of the straight line passing through (0,21) and (3,0) is:

By substituting the respective values in the above equation of straight line (1) we get,

\[\left( y-{{y}_{1}} \right)=\left( \dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}} \right)\left(
x-{{x}_{1}} \right)\]

\[\Rightarrow \left( y-21 \right)=\left( \dfrac{0-21}{3-0} \right)\left( x-0 \right)\]
\[\Rightarrow \left( y-21 \right)=\left( \dfrac{-21}{3} \right)x\]

On further simplification we get,

\[\Rightarrow \left( y-21 \right)=-7x\]

Now, take all the terms to the left hand side.

\[\therefore 7x+y=21\]

We already know that we require the points which lie strictly inside the triangle.

Let us assume the coordinate of the required point be (x, y).

By observing the diagram we can say that the point that should be inside the triangle will be on the same side of the origin with respect to the equation of the line we got.

A point (x, y) will lie on the side of the origin relative to a line \[a{{x}_{1}}+b{{y}_{1}}+c=0\], if \[ax+by+c=0\] and \[c\] have the same sign.

We have the equation of line as \[7x+y-21=0\].

Here, \[c=-21\] which is less than 0. So,

\[7x+y-21<0\]

\[7x+y<21\]

Now, the possible values of x and y that satisfy the above inequality are:

(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (1,7), (1,8), (1,9), (1,10), (1,11), (1,12), (1,13) .

These are the integral points that lie strictly inside the triangle.

Hence, there are 13 integral coordinates that lie strictly inside the given triangle.

Note: Instead of using the equation of line passing through (0,21) and (3,0) we can also use the lines passing through (0,0) and (3,0) and also line passing through (0,0) and (0,21) and follow the same process. In the other way also we can find the equation of the line and then consider some other point that should be inside the triangle because we cannot consider the origin in this case. It is a bit confusing to solve that way. More points will be included if we consider the points that lie on the triangle also which are the points that satisfy the given condition of equality. But, as mentioned here strictly inside, we did not consider the points that lie on the triangle