MathematicsChapter 7: Coordinate Geometry
Midpoint in Figures
Three standard figures use the midpoint.
Learning objectives
- Explain midpoint in figures in clear mathematical language
- Apply the ideas in parallelogram and circle
- Solve problems involving mid-segment of a triangle
Definition
Three standard figures use the midpoint. In a parallelogram the diagonals bisect each other, so they share midpoint; that is a coordinate test for parallelogram. In a circle the centre is the midpoint of diameter AB. In a triangle the mid-segment (the join of the midpoints of two sides) is parallel to the third side and half as long. Each claim is a midpoint calculation plus a distance or slope check.
Where / symbols
- parallelogram
- In a parallelogram the diagonals bisect each other, so they share midpoint; that is a coordinate test for parallelogram.
- share midpoint
- In a parallelogram the diagonals bisect each other, so they share midpoint; that is a coordinate test for parallelogram.
- centre
- In a circle the centre is the midpoint of diameter AB.
- midpoint of diameter
- In a circle the centre is the midpoint of diameter AB.
Overview
Same formula, three pictures:
Parallelogram and Circle
For quadrilateral ABCD to be a parallelogram, the diagonals AC and BD must share a midpoint: midpoint of A,C equals midpoint of B,D. That single check is often quicker than comparing all four sides. For a circle, if AB is a diameter then the centre is the midpoint of A and B. Every other radius will match the distance from that centre to A.
Examples Diagonals of a parallelogram meet at their midpoints · Diameter ends (2,1) and (6,5), centre (4,3) · Equal diagonal midpoints => parallelogram
Mid-Segment of A Triangle
Join the midpoints of two sides of a triangle. That mid-segment is parallel to the third side and half as long. If M and N are midpoints of AB and AC, then MN is parallel to BC and MN = (12) BC. Prove it with midpoint coordinates, then distance (half) and, in the next chapter, equal slopes (parallel).
Examples Midpoints of AB and AC join parallel to BC · MN = (1/2) BC · A(0,0), B(4,0), C(0,6): mid-segment length 3
What it means
Three standard figures use the midpoint.
Main ideas
1. PARALLELOGRAM AND CIRCLE: For quadrilateral ABCD to be a parallelogram, the diagonals AC and BD must share a midpoint: midpoint of A,C equals midpoint of B,D. 2. MID-SEGMENT OF A TRIANGLE: Join the midpoints of two sides of a triangle.
Why it matters
Always name which segment you mid-pointed. A parallelogram test uses both diagonals. A circle centre uses a diameter, not a random chord (the midpoint of a chord is not the centre unless the chord is a diameter). A mid-segment needs two midpoints, not a midpoint and a vertex.
How to answer in an exam
Parallelogram: diagonals share midpoint. Circle: centre = midpoint of diameter. Triangle mid-segment: parallel to the third side and half as long. Do not use the midpoint of a chord as a centre unless the question says the chord is a diameter.
Explanation
Always name which segment you mid-pointed. A parallelogram test uses both diagonals. A circle centre uses a diameter, not a random chord (the midpoint of a chord is not the centre unless the chord is a diameter). A mid-segment needs two midpoints, not a midpoint and a vertex.
In simple words
Parallelogram: diagonals share midpoint. Circle: centre = midpoint of diameter. Triangle mid-segment: parallel to the third side and half as long. Do not use the midpoint of a chord as a centre unless the question says the chord is a diameter.
Key points
- parallelogram
- share midpoint
- centre
- midpoint of diameter
- share a midpoint
- diameter
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Midpoint in Figures
Sticky Notes · Quick Revision
Core Idea
- Three standard figures use the midpoint.
Key Terms
- parallelogram: In a parallelogram the diagonals bisect each other, so they share midpoint; that is a coordinate test for parallelogram.
- share midpoint: In a parallelogram the diagonals bisect each other, so they share midpoint; that is a coordinate test for parallelogram.
Main Pattern
- 1. PARALLELOGRAM AND CIRCLE: For quadrilateral ABCD to be a parallelogram, the diagonals AC and BD must share a midpoint: midpoint of A,C equals midpoint of B,D.
- 2. MID-SEGMENT OF A TRIANGLE: Join the midpoints of two sides of a triangle.
Worked Reminder
- Diagonals of a parallelogram meet at their midpoints
- Diameter ends (2,1) and (6,5), centre (4,3)
- Equal diagonal midpoints => parallelogram
Common Mistake
- Parallelogram: diagonals share midpoint. Circle: centre = midpoint of diameter. Triangle mid-segment: parallel to the third side and half as long. Do not use the midpoint of a chord as a centre unless the question says the chord is a diameter.
Exam Tip
- State the rule for midpoint in figures, show each algebraic step, and check the final answer against the question.
60-second revision
- Three standard figures use the midpoint.
- For quadrilateral ABCD to be a parallelogram, the diagonals AC and BD must share a midpoint: midpoint of A,C equals midpoint of B,D.
- Join the midpoints of two sides of a triangle.
- Always name which segment you mid-pointed.
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