Ratios and parallelograms

When the ratio is what you're asked for, call it k : 1. One unknown is easier than two. Then:

P = ( (kx₂ + x₁)/(k + 1) , (ky₂ + y₁)/(k + 1) )

Use one coordinate to find k, then check with the other. For (–4, 6) on A(–6, 10)B(3, –8): (3k – 6)/(k + 1) = –4 gives k = 2/7. So the ratio is 2 : 7, and the y-coordinate checks out.

'An axis divides the segment' tells you one coordinate of the point. A point on the y-axis has x = 0, and a point on the x-axis has y = 0. The y-axis cuts (5, –6) to (–1, –4) where (–k + 5)/(k + 1) = 0, so k = 5. The ratio is 5 : 1, at (0, –13/3).

A missing corner of a parallelogram. The diagonals of a parallelogram cut each other in half, so both diagonals have the same midpoint. Write that reason down, then set the two midpoints equal.

The Sports Day ground (from the book): Niharika's green flag is at (2, 25) and Preet's red flag at (8, 20). Rashmi puts a blue flag exactly halfway, at the midpoint (5, 22.5).

Equidistant is not the same as midpoint. A point can be the same distance from I and J without being on the line IJ at all. Only a point that is equidistant and on the line is the midpoint.

Area of a rhombus = ½ × (one diagonal) × (the other). Find each diagonal with the distance formula.