Question

#MODS challenge
Open challenge: [Coordinate geometry]
A triangle is formed by the points (0,0), (8,0) and (0,8). How many points with integral coordinates are in the interior of the triangle.
[Ans. 21]

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Answer 1

Question :

A triangle is formed by the points (0,0), (8,0) and (0,8). How many points with integral coordinates are in the interior of the triangle

Solution :

We can solve this problem in two ways

1) Graphical approach

2) Direct formula

I'm solving by Graphical approach :

Let, A (0,0) , B (8,0) , C ( 0,8) be three coordinates of triangle take a point D ( 8 , 8 ) on the plane. and join ABDC which will give a square

Total number of points associated with square = 9 × 9 = 81

No. of points lying on boundary of square

= 2 × 16 = 32

So, remaining points lying inside the square = 81 - 32 = 49

Now, number of points lying on hypotenuse of triangle ABC and inside the square = 7

So,no. of points remaining inside the square on removing points lying on hypotenuse of triangle ABC = 49 - 7 = 42

Since, 42 points lie inside the square (on removing points lying on hypotenuse of ABC)

so, Points lying inside triangle ABC = 42/2 = 21

Therefore, 21 points with integral coordinates are in the interior of the triangle.

Answer 2

$\large\bf\blue{Question➠}$

A triangle is formed by the points (0,0), (8,0) and (0,8). How many points with integral coordinates are in the interior of the triangle.

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$\large\bf\green{Answer➠}$

Total Integral Coordinates are$\bf\red{➠21 }$

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$\large\bf\pink{SOLUTION➠}$

(SEE THE ATTACHMENT ABOVE)

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$\bf\orange{Note:-}$

There are three ways to solved this problem

1).Direct Formula

2).Graphical illustration

3).Equation formula

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$\large\bf\purple{Formula\: used➽}$

3). Equation Formula.

(Creating Equations of all line segments and then find the integral point by there total sum)