In the coordinate plane, a point is called a lattice point if both of its coordinates are integers. Let A be the point (12,84) . Find the number of right angled triangles A B C in the coordinate plane where B and C are

Question

Quantrex Academy · Accepted Answer

Correct answer: 24

We know that if inradius is integer then sides of triangle is also integer. In the coordinate plane, a point is called a lattice point if both of its coordinates are integers. Let A be the point

In the coordinate plane, a point is called a lattice point if both of its coordinates are integers. Let A be the point

Here

O A=60 \sqrt{2}

. Inradius

=60

Hence

A B, B C, C A

are integer. Let

A B=m^2-n^2

B C=m^2+n^2

and

A C=2 m^2+n^2

Here,

r=\frac{\Delta}{s}=\frac{m n \cdot\left(m^2-n^2\right)}{3(m+n)}=n(m-n)=60

\therefore n(m-n)=2^2 .3 .5 .

\therefore

Possible values of

n

are 12 . Then 12 such triangles are possible but when

B

and

C

interchange their position 12 more triangles are possible. Then total number of distinct triangles

=24

.

In the coordinate plane, a point is called a lattice point if both...

Correct answer

Step-by-step solution

More from Straight Lines