P is one-third and Q is two-thirds along the segment. The total change in x is 12 and y is –9. Applying these shifts to A gives P (8, 4) and Q (12, 1).
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(i) Right-angled isosceles: O (0, 0), A (5, 0), B (0, 5). (ii) Isosceles: O (0, 0), P (–3, –4) in Quadrant III and Q (3, –4) in Quadrant IV. Both triangles have two equal side lengths.
Check the change in y over the change in x. From R to B: (–5 – –1) / (–2 – –5) = –4/3. From B to C: (–12 – –5) / (4 – –2) = –7/6. Slopes differ, so they ...
Yes, they are collinear. One method is comparing the ratio of coordinates (y/x). For M, –4 divided by –3 equals 4/3; for G, 8 divided by 6 equals 4/3. Since ratios match, they are on one line.
Without negative numbers, the system would only include Quadrant I, where both coordinates are positive. This restricted system would not allow us to locate points in the other three quadrants of a full 2-D plane.