9. We would normally use point-point form to determine three points on a line. Here A and C have the same y coordinate, so we skip all that and just write K=3.
10. Perimeters given coordinates can get messy because of the square roots
i. A(3,10), B(5,2), C(4,12)
AB² = (5-3)² + (2-10)² = 68 = 2²·17
BC²= (4-5)² + (12-2)² = 101
AC² = (4-3)² + (12-10)² = 5
Perimeter: 2√17 + √101 + √5
ii. A(-2,1), B(4,6), C(6,3)
AB² = (4 - - 2)² + (6-1)² = 61
BC²= (6-4)² + (3-6)² = 13
AC² = (6 - -2)² + (3-1)² = 68
Perimeter: √61 + √13 + 2√17
I'll leave any calculator approximating to you; I prefer exact answers.
11.
When the triangles have a common x or y coordinate we can just do half base times height. The first and the third question here don't work that way. We can use the Shoelace Formula which says the signed area is half the sum of the cross products of the sides, true for any polygon.
Actually let's translate the first vertex to the origin because the area of triangle with vertices (0,0), (a,b), (c,d) is (1/2)(ad - bc)
i. Translating (1,1), (-4,6), (-3,-5) becomes (0,0), (-5,5), (-4,-6) so area
(1/2)( -5(-6) - 5(-4) ) = (1/2)(50) = 25
Answer: 25
ii. We have common coordinate, one side parallel to an axis, which makes things easier. We have a base along x=2 of length b=6-3=3. There's an altitude h=4 - 2 = 2, so area is (1/2)(3)(2) = 3.
iii. (2,-2), (-2,1) & (5,2)
No common coordinate. Let's try the full shoelace:
Δ = (1/2)( 2(1) - -2(-2) + (-2)(2) - 1(5) + 5(-2) - 2(2) ) = -25/2
The Shoelace Formula gives signed area, which may be negative. The area is the absolute value: 25/2