Explanation:
for the lengths of BC and AC we can count the squares, because the lengths of these lines are the sum of the side lengths of each inducing square.
1.
BC = 7 - -2 = 7 + 2 = 9 cm
2.
AC = 13 - 1 = 12 cm
3.
but for AB we cannot do that, because AB consists of the lengths of all the diagonals of the squares. and a diagonal must always be longer than any side.
just look at the triangle : it is clear that AB must be longer than AC.
no, for a line that goes across the grid squares we need other methods.
in our case, we have a right-angled triangle and can use Pythagoras
c² = a² + b²
c being the Hypotenuse (the side opposite of the 90° angle), a and b being the legs of the triangle (enclosing the 90° angle).
AB is in our case the Hypotenuse, and we get
AB² = 12² + 9² = 144 + 81 = 225
AB = sqrt(225) = 15 cm
4.
as explained, we used Pythagoras.
5.
the coordinates for a midpoint between 2 points (x1, y1) and (x2, y2) are simply
((x1+x2)/2, (y1+y2)/2)
A = (13, -2)
B = (1, 7)
C = (1, -2)
midpoint AC = ((13+1)/2, (-2 + -2)/2) = (14/2, -4/2) =
= (7, -2)
the x coordinate is 7.
6.
midpoint BC = ((1+1)/2, (7+ -2)/2) = (2/2, 5/2) = (1, 2.5)
the y coordinate is 2.5
7.
midpoint AB = ((13+1)/2, (-2+7)/2) = (14/2, 5/2) =
= (7, 2.5)