Question

The number of integral points (integral point means both the coordinates should be an integer) exactly in the interior of the triangle with vertices $\left(0,0\right),(0,21)$, and $\left(21,0\right)$, is

• A. $133$
• B. $190$
• C. $233$
• D. $105$

Answer 1

The correct option is B

$190$

Explanation of the correct option.

Step 1: Visually represent the given data.

Draw the given triangle and assume that, $O=\left(0,0\right),Q=(0,21)$ and $P=\left(21,0\right)$.

Now, find the slope $m$ of the line $PQ$ as follows:

$$\begin{gathered} m=\frac{0-21}{21-0} \\ \Rightarrow m=-1 \end{gathered}$$

Since the slope of the line $PQ$ is $-1$ which means that if we increase the value of $x$ by $1$ there is a decrease in the value of $y$ by $1$.

Step 2: Find the number of required points.

We have to find the number of integral points exactly in the interior of the given triangle.

For $x=1$.

The coordinates of $y$ that lie exactly inside the given triangle are $(1,1),(1,2),(1,3),(1,4),...,(1,19)$.

So, the number of points that lie exactly inside the given triangle for $x=1$ is $19$.

For $x=2$.

The coordinates of $y$ that lie exactly inside the given triangle are $(2,1),(2,2),(2,3),(2,4),...,(2,18)$.

So, the number of points that lie exactly inside the given triangle for $x=2$ is $18$ and so on.

And for $x=20$.

The only point which lies inside the triangle is $(20,0)$.

Find the total number of points that lie exactly inside the given triangle can be given by:

$$\begin{gathered} 19+18+....+1=\frac{19}{2}(2·19+(19-1\left)\right(-1\left)\right) \\ \Rightarrow 19+18+....+1=\frac{19}{2}\left(20\right) \\ \Rightarrow 19+18+....+1=190 \end{gathered}$$

Therefore, the total number of points that lies exactly inside the given triangle is $190$.

Hence, option $B$ is the correct option.