IPMAT Indore 2019 (MCQ) - The number of points, having both coordinates as integers, that lie in the interior of the triangle with vertices (0, 0), (0, 31), and (31, 0) is | PYQs + Solutions

IPMAT Indore 2019

Geometry

Triangles

Medium

The number of points, having both coordinates as integers, that lie in the interior of the triangle with vertices $(0, 0), (0, 31),$ and $(31, 0)$ is

435

465

450

464

Correct Option: 1

The question requires us to find the number of points that lie 'inside' the triangle formed by the vertices

(0,0), (0,31)

and

(31,0)

The image below shows the graph of the triangle formed by the three given coordinates.

Since we need to find the number of points that lie inside the triangle, we must NOT consider the points that lie on the boundary.

Starting with

(1,1)

(1,29)

, there are 29 points.

\newline

Similarly, from

(2,1)

(2,28)

, there are 28 points, from

(3,1)

(3, 27)

there are 27 points and so on.

\newline

At last, there is only one coordinate

(29,1)

before

(31,0)

that lies inside the triangle.

Hence, the total number of integer-coordinate-points that lie inside the triangle are:

\newline

29+28+27+ \dots + 1

Using the formula of sum of first

n

-natural numbers,

\dfrac{n(n+1)}{2} = \dfrac{29 \times 30}{2} = 435

Therefore, the answer is

435