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Coordinates with Ratios
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Foundational Skills
Understand ratio on a line segment
\[ 2:3 \rightarrow \frac{2}{5} \text{ of way} \]
Convert ratio notation to a fraction of the total distance.
Find total parts from a ratio
\[ 3:5 \rightarrow 8 \text{ parts} \]
Count total parts in a ratio.
Interpret ratio direction
\[ 1:4 \text{ from A} \rightarrow 4:1 \text{ from B} \]
Understand that ratio direction matters.
Recognise midpoint as equal ratio
\[ \text{Midpoint} = 1:1 \]
Identify that midpoint corresponds to ratio 1:1.
Midpoint Calculations
Find midpoint coordinates
\[ M = \left( \frac{x_1+x_2}{2}, \frac{y_1+y_2}{2} \right) \]
Calculate the midpoint of a line segment.
Midpoint with negatives
\[ (-4,6), (2,-2) \rightarrow (-1,2) \]
Handle negative coordinates in midpoint calculation.
Find endpoint given midpoint
\[ B = 2M – A \]
Work backwards from midpoint to find an endpoint.
Verify a midpoint
\[ \text{Is } (5,3) \text{ midpoint?} \]
Check if a given point is the midpoint.
Section Formula
Point dividing in ratio 1:2
\[ 1:2 \rightarrow \frac{1}{3} \text{ of way} \]
Apply section formula with ratio 1:2.
Point dividing in ratio 2:1
\[ 2:1 \rightarrow \frac{2}{3} \text{ of way} \]
Apply section formula with ratio 2:1.
Point dividing in ratio 1:3
\[ 1:3 \rightarrow \frac{1}{4} \text{ of way} \]
Apply section formula with ratio 1:3.
Point dividing in ratio 3:1
\[ 3:1 \rightarrow \frac{3}{4} \text{ of way} \]
Apply section formula with ratio 3:1.
Point dividing in ratio 2:3
\[ 2:3 \rightarrow \frac{2}{5} \text{ of way} \]
Apply section formula with ratio 2:3.
Section formula with negatives
\[ (-6,4) \text{ to } (6,-2) \]
Handle negative coordinates in section formula.
Working Backwards
Find ratio given dividing point
\[ A, B, P \rightarrow AP:PB \]
Determine the ratio from coordinates of the dividing point.
Find ratio from midpoint check
\[ \text{Verify } AM:MB \]
Verify if a point is the midpoint and state the ratio.
Find endpoint given ratio
\[ P, A, \text{ratio} \rightarrow B \]
Work backwards from the ratio to find an unknown endpoint.
Single Coordinate
Find x-coordinate of dividing point
\[ \text{Find } P_x \]
Apply ratio to single coordinate.
Find y-coordinate of dividing point
\[ \text{Find } P_y \]
Apply ratio to y-coordinate only.
Applications
Find trisection points
\[ \text{Ratios } 1:2 \text{ and } 2:1 \]
Find both trisection points (ratios 1:2 and 2:1).
Find quarter points
\[ \frac{3}{4} \text{ of the way} \]
Find a point at a fractional distance.
Centroid of triangle
\[ G = \left( \frac{x_1+x_2+x_3}{3}, … \right) \]
Calculate centroid as average of vertices.
Point at given distance ratio
\[ \frac{2}{5} \text{ of journey} \]
Apply ratio in a practical context.
Problem Solving
Verify point on line segment
\[ \text{Does P lie on AB?} \]
Check if a point divides a segment in a valid ratio.
Find point not on segment
\[ \text{Ratios don’t match} \]
Identify when ratios don’t match (point not on segment).
Intersection with axis
\[ \text{Where line crosses x-axis} \]
Find where a line crosses an axis using ratio.
Special Cases
External division
\[ P = 2B – A \]
Understand external division (point outside segment).
Symmetric point about midpoint
\[ B = 2M – A \]
Find the reflection of a point about the midpoint.
Timer (Optional)
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