Question

Coordinate Geometry

Answer 1

The number of points, having both co-ordinates as integers , that lie in the interior of the triangle with vertices

\(\large \color{black }{\begin{align}
(0,0),(0,41)\quad and \quad (41,0)\hspace{.33em}\\~\\
\end{align}}\)

is

Answer 2

\(\large \color{black }{\begin{align}
&a.)\quad 820\hspace{.33em}\\~\\
&b.)\quad 780\hspace{.33em}\\~\\
&c.)\quad 901\hspace{.33em}\\~\\
&d.)\quad 861\hspace{.33em}\\~\\
\end{align}}\)

Answer 3

this = babaies

Answer 4

the slope of the hypotenuse is -1, so the "tops" of each column are integers 41,40,... ,0
if we start at x=1, the entire column has its top is at 40
the number of inside points are y=1 to 39

at x=2, its top is at 39, # of inside points are y=1 to 38
etc
add up 39,38,37, down to 1

Answer 5

http://www.wolframalpha.com/input/?i=%5Csum%5Climits_%7Bk%3D1%7D%5E%28p-1%29+%28k-1%29

plugin p = 41

Answer 6

i just drew by hand and followed a pattern

\(\large \color{black }{\begin{align}
(0,0),(0,1),(1,0)\implies 0\hspace{.33em}\\~\\
(0,0),(0,2),(2,0)\implies 0\hspace{.33em}\\~\\
(0,0),(0,3),(3,0)\implies 1\hspace{.33em}\\~\\
(0,0),(0,4),(4,0)\implies 3\hspace{.33em}\\~\\
(0,0),(0,5),(5,0)\implies 6\hspace{.33em}\\~\\
(0,0),(0,6),(6,0)\implies 10\hspace{.33em}\\~\\
\cdots \text{which gives triangular numbers}

\end{align}}\)

\(\large \color{black }{\begin{align} \dfrac{(41-2)(41-2+1)}{2}\end{align}}\)