Coordinates: distance formula, slope formula, point slope formula, section formula, prove perpendicular, prove collinear; circumcenter, centroid, orthocenter. And some practice questions different from very beginning to so hard and complex. Hong Kong DSE style.
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The distance formula is fundamental in coordinate geometry. To find the distance between two points, we use the Pythagorean theorem. For points A at (1,1) and B at (4,3), we form a right triangle. The horizontal distance is 3 units, the vertical distance is 2 units, giving us a distance of square root of 13.
The slope formula measures how steep a line is. It's calculated as rise over run, or the change in y divided by the change in x. For points C at (1,1) and D at (4,3), the rise is 2 and the run is 3, giving a slope of two-thirds. When two lines are perpendicular, the product of their slopes equals negative one.
The point-slope formula helps us write equations of lines when we know a point and the slope. The section formula finds coordinates of points that divide line segments in specific ratios. Point P divides segment AB in ratio 1 to 1, making it the midpoint. For internal division, we use the weighted average of coordinates.
To prove lines are perpendicular, we check if the product of their slopes equals negative one. Here, line one has slope 2 and line two has slope negative one-half. Their product is negative one, confirming perpendicularity. For collinearity, points A, B, and C all lie on the same line with slope one-half.
Triangle centers are special points with unique properties. The circumcenter is where perpendicular bisectors meet, equidistant from all vertices. The centroid, at coordinates (3, 2), is where medians intersect and represents the triangle's center of mass. The orthocenter is where altitudes meet. These centers are fundamental in triangle geometry and coordinate proofs.