Question 13: Coordinate Geometry of a Kite

Given B = (−1, 4), D = (2, 1), A is on the x-axis, AC ⟂ BD, M is the midpoint of BD, and AC = 5/2 AM. Find the equation of AC, A, C, and the area of kite ABCD.

Diagonal BD Diagonal AC Point M Kite ABCD
Visualise AC = k × AM
Move the slider to see why the final value k = 5/2 places C beyond M on the same straight line.
Current value: AC = 2.5 × AM

Step 1

Foundation
Foundation idea 1: midpoint
The midpoint is found by averaging the x-coordinates and averaging the y-coordinates.
M = ((x₁ + x₂)/2, (y₁ + y₂)/2)
Foundation idea 2: gradient
Gradient means “change in y divided by change in x”.
gradient = (y₂ − y₁)/(x₂ − x₁)
Foundation idea 3: perpendicular lines
If two lines are perpendicular, their gradients multiply to −1. For example, if one gradient is −1, the perpendicular gradient is 1.
Foundation idea 4: area of kite
When the diagonals are perpendicular:
Area = 1/2 × diagonal 1 × diagonal 2
Quick check
Final answers
Equation of AC: y = x + 2
A: (−2, 0)
C: (17/4, 25/4) or (4.25, 6.25)
Area: 75/4 square units