7.3 Motion and Construction
Moving and Changing Shapes
Transformations are ways to move or change shapes on a coordinate grid. There are four main types you need to know.
Transformations are ways to move or change shapes on a coordinate grid. There are four main types you need to know.
⚡ The Four Transformations:
Translation
Sliding a shape
Same size, same orientation
Same size, same orientation
Rotation
Turning around a point
Reflection
Flipping over a line
Enlargement
Changing size from a point
Example 1: Translation
A triangle has vertices at (1, 2), (3, 2), and (1, 4). It is translated by the vector $\begin{pmatrix}4\\-1\end{pmatrix}$. Find the new coordinates.
Solution:
Add the vector to each coordinate:
• $(1, 2) + (4, -1) = (5, 1)$
• $(3, 2) + (4, -1) = (7, 1)$
• $(1, 4) + (4, -1) = (5, 3)$
Answer: New vertices are (5,1), (7,1), and (5,3)
Add the vector to each coordinate:
• $(1, 2) + (4, -1) = (5, 1)$
• $(3, 2) + (4, -1) = (7, 1)$
• $(1, 4) + (4, -1) = (5, 3)$
Answer: New vertices are (5,1), (7,1), and (5,3)
Example 2: Rotation
Rotate the point (3, 1) by 90° clockwise about the origin (0, 0).
Rule for 90° clockwise about origin:
$(x, y) \rightarrow (y, -x)$
Solution:
$(3, 1) \rightarrow (1, -3)$
Answer: The new coordinates are (1, -3)
• 90° anticlockwise: $(x, y) \rightarrow (-y, x)$
• 180°: $(x, y) \rightarrow (-x, -y)$
Rule for 90° clockwise about origin:
$(x, y) \rightarrow (y, -x)$
Solution:
$(3, 1) \rightarrow (1, -3)$
Answer: The new coordinates are (1, -3)
Rotation Rules (about origin):
• 90° clockwise: $(x, y) \rightarrow (y, -x)$• 90° anticlockwise: $(x, y) \rightarrow (-y, x)$
• 180°: $(x, y) \rightarrow (-x, -y)$
Example 3: Reflection
Reflect the point (4, 2) in the line $y = x$.
Rule for reflection in $y = x$:
Swap the coordinates
Solution:
$(4, 2) \rightarrow (2, 4)$
Answer: The reflected point is (2, 4)
• $y = -x$: $(x, y) \rightarrow (-y, -x)$
• $x$-axis: $(x, y) \rightarrow (x, -y)$
• $y$-axis: $(x, y) \rightarrow (-x, y)$
Rule for reflection in $y = x$:
Swap the coordinates
Solution:
$(4, 2) \rightarrow (2, 4)$
Answer: The reflected point is (2, 4)
Reflection Rules:
• $y = x$: $(x, y) \rightarrow (y, x)$• $y = -x$: $(x, y) \rightarrow (-y, -x)$
• $x$-axis: $(x, y) \rightarrow (x, -y)$
• $y$-axis: $(x, y) \rightarrow (-x, y)$
Example 4: Enlargement
Enlarge the point (2, 3) from center (0, 0) with scale factor 3.
Method:
Multiply each coordinate by the scale factor
Solution:
$(2, 3) \times 3 = (6, 9)$
Answer: The enlarged point is (6, 9)
Note: Scale factor > 1 makes it bigger
Scale factor between 0 and 1 makes it smaller
Method:
Multiply each coordinate by the scale factor
Solution:
$(2, 3) \times 3 = (6, 9)$
Answer: The enlarged point is (6, 9)
Note: Scale factor > 1 makes it bigger
Scale factor between 0 and 1 makes it smaller
💡 Quick Tips:
• Translation: Use vectors - direction and distance
• Rotation: Need center, angle, and direction (clockwise/anticlockwise)
• Reflection: Need the mirror line
• Enlargement: Need center and scale factor
Always mark the center of rotation/enlargement with a cross!
• Rotation: Need center, angle, and direction (clockwise/anticlockwise)
• Reflection: Need the mirror line
• Enlargement: Need center and scale factor
Always mark the center of rotation/enlargement with a cross!
🎯 Identify the Transformation
Understanding Bearings
A bearing is a way of describing direction using angles. It's used in navigation for ships, planes, and hiking.
A bearing is a way of describing direction using angles. It's used in navigation for ships, planes, and hiking.
⚡ The Three Golden Rules for Bearings:
1. From North
2. Clockwise
3. Three Digits
Common Bearings:
Example 1: Finding a Bearing
Town A is directly East of Town B. What is the bearing from B to A?
Solution:
• Start from North at B
• Turn clockwise to point towards A
• East is 90° from North
Answer: The bearing is 090°
• Start from North at B
• Turn clockwise to point towards A
• East is 90° from North
Answer: The bearing is 090°
Example 2: Back Bearings
The bearing from A to B is 065°. Find the bearing from B to A.
Rule: Back bearing differs by 180°
• If bearing < 180°: add 180°
• If bearing > 180°: subtract 180°
Solution:
$065° < 180°$, so add 180°
$065° + 180° = 245°$
Answer: The bearing from B to A is 245°
• If bearing < 180°: add 180°
• If bearing > 180°: subtract 180°
Solution:
$065° < 180°$, so add 180°
$065° + 180° = 245°$
Answer: The bearing from B to A is 245°
Example 3: Scale Drawing Problem
A ship sails on a bearing of 120° for 50 km, then on a bearing of 210° for 30 km. How far is it from the starting point?
Method:
1. Draw accurate diagram using scale (e.g., 1 cm = 10 km)
2. Use protractor for bearings
3. Measure distance on diagram
4. Convert back using scale
Note: This is best done with proper drawing equipment!
Method:
1. Draw accurate diagram using scale (e.g., 1 cm = 10 km)
2. Use protractor for bearings
3. Measure distance on diagram
4. Convert back using scale
Note: This is best done with proper drawing equipment!
💡 Bearing Tips:
• Always draw a North line at each point
• Use a protractor to measure angles
• Remember: North = 000°, East = 090°, South = 180°, West = 270°
• Write bearings with 3 digits (045° not 45°)
• Use a protractor to measure angles
• Remember: North = 000°, East = 090°, South = 180°, West = 270°
• Write bearings with 3 digits (045° not 45°)
🎯 Calculate Back Bearings
°
°
What is a Locus?
A locusis a set of points that satisfy a particular condition or rule. It creates a path or region.
A locusis a set of points that satisfy a particular condition or rule. It creates a path or region.
⚡ Common Loci:
Fixed Distance from Point
Locus = Circle
All points distance $d$ from point P
Equidistant from Two Points
Locus = Perpendicular Bisector
Equal distance from A and B
Equidistant from Two Lines
Locus = Angle Bisector
Equal distance from both lines
Fixed Distance from Line
Locus = Parallel Lines
Distance $d$ from line L
Example 1: Circle Locus
Draw the locus of points exactly 3 cm from point X.
Method:
1. Mark point X
2. Set compass to 3 cm
3. Place compass point on X
4. Draw a complete circle
Result: A circle with center X and radius 3 cm
Method:
1. Mark point X
2. Set compass to 3 cm
3. Place compass point on X
4. Draw a complete circle
Result: A circle with center X and radius 3 cm
Example 2: Perpendicular Bisector
Points A and B are 6 cm apart. Construct the locus of points equidistant from A and B.
Method:
1. Open compass to more than half the distance AB
2. Place compass on A, draw arcs above and below
3. Keep same compass width, place on B, draw arcs
4. Join the two intersection points with a straight line
Result: The perpendicular bisector of AB
Method:
1. Open compass to more than half the distance AB
2. Place compass on A, draw arcs above and below
3. Keep same compass width, place on B, draw arcs
4. Join the two intersection points with a straight line
Example 3: Angle Bisector
Construct the locus of points equidistant from two lines that meet at point O.
Method:
1. From O, mark equal points P and Q on each line
2. From P, draw an arc
3. From Q, draw an arc with same radius
4. Join O to where the arcs cross
Result: The angle bisector dividing the angle in half
Method:
1. From O, mark equal points P and Q on each line
2. From P, draw an arc
3. From Q, draw an arc with same radius
4. Join O to where the arcs cross
Result: The angle bisector dividing the angle in half
💡 Construction Tips:
• Use a sharp pencil for accuracy
• Don't change compass width during construction
• Draw construction lines lightly
• For angle bisector: same radius from each line
• Label all important points clearly
• Don't change compass width during construction
• Draw construction lines lightly
• For angle bisector: same radius from each line
• Label all important points clearly
Real Life Uses:
• Mobile Phone Masts: Loci show coverage areas
• Emergency Services: Finding locations equidistant from stations
• Archaeology: Defining search areas
• Sports Fields: Marking boundaries at fixed distances
• Engineering: Designing curves and paths
• Navigation: Bearings for ships and aircraft
• Mobile Phone Masts: Loci show coverage areas
• Emergency Services: Finding locations equidistant from stations
• Archaeology: Defining search areas
• Sports Fields: Marking boundaries at fixed distances
• Engineering: Designing curves and paths
• Navigation: Bearings for ships and aircraft
🎯 Identify the Locus