Mathematics · Topic 1.3
Transformations & Pythagoras
Learn to move and resize shapes on the coordinate plane using four types of transformations. Then discover how Pythagoras' theorem connects the sides of every right-angled triangle.
What You'll Learn
- Perform translations, reflections, rotations, and enlargements on the coordinate plane
- Describe transformations precisely using mathematical language
- Understand scale factor and its effect on shape size
- State and apply Pythagoras' theorem to find missing sides
- Use Pythagoras' theorem in real-world contexts (distances, diagonals)
IB Assessment Focus
Criterion A: Apply transformation rules and Pythagoras' theorem accurately.
Criterion B: Investigate patterns — e.g., what stays the same and what changes under each transformation.
Criterion C: Use correct geometric notation and describe transformations precisely.
Criterion D: Apply Pythagoras' theorem to real-world measurement problems.
Key Vocabulary
| Term | Definition |
|---|---|
| Transformation | A change in the position, size, or orientation of a shape |
| Image | The new shape after a transformation (labelled with prime notation: A′) |
| Congruent | Shapes that are identical in size and shape (translations, reflections, rotations produce congruent images) |
| Similar | Shapes with the same angles but different sizes (enlargements produce similar images) |
| Hypotenuse | The longest side of a right-angled triangle; opposite the right angle |
Mathematics · Topic 1.3
Translations, Reflections & Rotations
Three transformations that move a shape without changing its size. The image is always congruent to the original.
Translation (Slide)
- Every point moves the same distance in the same direction
- Described by a vector: e.g., (+3, −2) means "right 3, down 2"
- Shape, size, and orientation stay the same — only position changes
Example: Point A(2, 5) translated by (+3, −2) → A′(2+3, 5+(−2)) = A′(5, 3)
Reflection (Flip)
- Every point is mirrored over a line of reflection
- Common lines: x-axis, y-axis, y = x, or any given line
- Shape and size stay the same, but orientation is reversed (flipped)
| Reflection Line | Rule | Example |
|---|---|---|
| x-axis | (x, y) → (x, −y) | (3, 4) → (3, −4) |
| y-axis | (x, y) → (−x, y) | (3, 4) → (−3, 4) |
| y = x | (x, y) → (y, x) | (3, 4) → (4, 3) |
Rotation (Turn)
- Every point turns by a given angle around a centre of rotation
- You must state: angle, direction (clockwise/anticlockwise), and centre point
- Shape and size stay the same; position and orientation change
| Rotation (about origin) | Rule | Example |
|---|---|---|
| 90° anticlockwise | (x, y) → (−y, x) | (3, 4) → (−4, 3) |
| 180° | (x, y) → (−x, −y) | (3, 4) → (−3, −4) |
| 90° clockwise | (x, y) → (y, −x) | (3, 4) → (4, −3) |
Mathematics · Topic 1.3
Enlargement & Scale Factor
Enlargement changes the size of a shape while keeping its proportions the same. The image is similar to the original.
Scale Factor
New length = Original length × Scale factor
- Scale factor > 1: Shape gets bigger
- 0 < Scale factor < 1: Shape gets smaller
- Scale factor = 1: No change (same size)
- Angles never change in an enlargement
- The image is similar to the original (same shape, different size)
Example: A triangle with sides 3 cm, 4 cm, 5 cm is enlarged by scale factor 2.
New sides: 3 × 2 = 6 cm, 4 × 2 = 8 cm, 5 × 2 = 10 cm. All angles remain the same.
Transformations Summary
| Transformation | What stays the same | What changes | Image is… |
|---|---|---|---|
| Translation | Shape, size, orientation | Position | Congruent |
| Reflection | Shape, size | Position, orientation (flipped) | Congruent |
| Rotation | Shape, size | Position, orientation (turned) | Congruent |
| Enlargement | Shape, angles | Size and position | Similar |
Key Distinction: Translation, reflection, and rotation produce congruent images (same size). Enlargement produces a similar image (same shape but different size). Angles never change under any transformation.
Mathematics · Topic 1.3
Pythagoras' Theorem
One of the most important results in mathematics: in any right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.
Pythagoras' Theorem
a² + b² = c² where c is the hypotenuse (the longest side, opposite the right angle)
Finding the Hypotenuse
Example: Find c when a = 6 cm and b = 8 cm.
a² + b² = c²
6² + 8² = 36 + 64 = c²
c² = 100
c = √100 = 10 cm
Finding a Shorter Side
Rearrange the formula: a² = c² − b²
Example: Find a when c = 13 cm and b = 5 cm.
a² = c² − b²
a² = 13² − 5² = 169 − 25 = 144
a = √144 = 12 cm
Common Mistake: Pythagoras' theorem only works for right-angled triangles. Always check that there is a 90° angle before applying it. Also, c must always be the hypotenuse (the longest side).
Mathematics · Topic 1.3
Real-World Applications
Pythagoras' theorem appears in many practical situations: finding distances, diagonals, and heights.
Distance Between Two Points
To find the straight-line distance between two points on a coordinate plane, draw a right triangle and apply Pythagoras' theorem.
Example: Distance from A(1, 2) to B(4, 6).Find distance A(1,2) to B(4,6)
Horizontal distance: 4 − 1 = 3
Vertical distance: 6 − 2 = 4
Distance² = 3² + 4² = 9 + 16 = 25
Distance = √25 = 5 units
Diagonal of a Rectangle
Example: A rectangle measures 12 cm by 5 cm. Find the diagonal.
Diagonal = hypotenuse of right triangle with legs 12 and 5
d² = 12² + 5² = 144 + 25 = 169
d = √169 = 13 cm
Ladder Against a Wall
Example: A 10 m ladder leans against a wall, base 6 m from the wall. Height?
Ladder = hypotenuse (c = 10), base = b = 6
h² = 10² − 6² = 100 − 36 = 64
h = √64 = 8 m
Mathematics · Topic 1.3
Worked Examples
Study each multi-step solution carefully. Notice how transformations are described precisely.
EXAMPLE 1Describe the single transformation that maps A(2, 3) to A′(−2, 3).
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Full Solution
The y-coordinate stays the same (3) but the x-coordinate changes sign (2 → −2). This is a reflection in the y-axis. The rule (x, y) → (−x, y) confirms this: (2, 3) → (−2, 3). ✓
EXAMPLE 2Triangle PQR has vertices P(1,1), Q(4,1), R(4,3). Apply a translation of (−2, +3). Find the new vertices.
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Full Solution
Add the translation vector to each point:
P′ = (1+(−2), 1+3) = (−1, 4)
Q′ = (4+(−2), 1+3) = (2, 4)
R′ = (4+(−2), 3+3) = (2, 6)
The shape is unchanged — only its position has moved left 2 and up 3.
P′ = (1+(−2), 1+3) = (−1, 4)
Q′ = (4+(−2), 1+3) = (2, 4)
R′ = (4+(−2), 3+3) = (2, 6)
The shape is unchanged — only its position has moved left 2 and up 3.
EXAMPLE 3A triangle with sides 5 cm, 12 cm, and 13 cm. Verify that it is a right-angled triangle.
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Full Solution
If it is right-angled, then a² + b² should equal c² (where c is the longest side).
5² + 12² = 25 + 144 = 169
13² = 169
Since 5² + 12² = 13², the triangle is right-angled. ✓
The right angle is opposite the 13 cm side (the hypotenuse).
5² + 12² = 25 + 144 = 169
13² = 169
Since 5² + 12² = 13², the triangle is right-angled. ✓
The right angle is opposite the 13 cm side (the hypotenuse).
EXAMPLE 4A shape is enlarged by scale factor 3 from the origin. If vertex A is at (2, 4), find A′.
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Full Solution
For an enlargement from the origin with scale factor k, multiply each coordinate by k:
A′ = (2 × 3, 4 × 3) = (6, 12)
The new point is 3 times farther from the origin. All lengths in the shape are tripled, but all angles stay the same.
A′ = (2 × 3, 4 × 3) = (6, 12)
The new point is 3 times farther from the origin. All lengths in the shape are tripled, but all angles stay the same.
EXAMPLE 5Find the distance between the points (1, 1) and (7, 9).
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Full Solution
Horizontal distance: 7 − 1 = 6
Vertical distance: 9 − 1 = 8
Using Pythagoras: d² = 6² + 8² = 36 + 64 = 100
d = √100 = 10 units
Vertical distance: 9 − 1 = 8
Using Pythagoras: d² = 6² + 8² = 36 + 64 = 100
d = √100 = 10 units
EXAMPLE 6Point B(3, 2) is rotated 90° anticlockwise about the origin. Find B′.
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Full Solution
The rule for 90° anticlockwise rotation about the origin is (x, y) → (−y, x).
B(3, 2) → B′(−2, 3)
B′ = (−2, 3)
B(3, 2) → B′(−2, 3)
B′ = (−2, 3)
EXAMPLE 7A right triangle has hypotenuse 15 cm and one leg 9 cm. Find the other leg.
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Full Solution
Using the rearranged formula: a² = c² − b²
a² = 15² − 9² = 225 − 81 = 144
a = √144 = 12 cm
Check: 9² + 12² = 81 + 144 = 225 = 15² ✓
a² = 15² − 9² = 225 − 81 = 144
a = √144 = 12 cm
Check: 9² + 12² = 81 + 144 = 225 = 15² ✓
Mathematics · Topic 1.3
Practice Q&A
Attempt each question before revealing the model answer.
CALCULATEFind the hypotenuse of a right triangle with legs 8 cm and 15 cm.
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Model Answer
c² = 8² + 15² = 64 + 225 = 289
c = √289 = 17 cm
c = √289 = 17 cm
DESCRIBEDescribe the transformation that maps (5, 2) to (5, −2).
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Model Answer
The x-coordinate stays the same (5) and the y-coordinate changes sign (2 → −2). This is a reflection in the x-axis. The rule is (x, y) → (x, −y).
CALCULATEA shape has a side of 4 cm. After enlargement, the corresponding side is 12 cm. What is the scale factor?
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Model Answer
Scale factor = new length ÷ original length = 12 ÷ 4 = 3. The shape was enlarged by scale factor 3.
APPLYA TV screen is 80 cm wide and 60 cm tall. What is the diagonal measurement?
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Model Answer
d² = 80² + 60² = 6400 + 3600 = 10000
d = √10000 = 100 cm
d = √10000 = 100 cm
TRANSFORMPoint C(4, −1) is rotated 180° about the origin. Find C′.
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Model Answer
For 180° rotation about the origin: (x, y) → (−x, −y).
C(4, −1) → C′(−4, 1)
C′ = (−4, 1)
C(4, −1) → C′(−4, 1)
C′ = (−4, 1)
CALCULATEFind the missing side: hypotenuse = 25 cm, one leg = 7 cm.
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Model Answer
a² = 25² − 7² = 625 − 49 = 576
a = √576 = 24 cm
Check: 7² + 24² = 49 + 576 = 625 = 25² ✓
a = √576 = 24 cm
Check: 7² + 24² = 49 + 576 = 625 = 25² ✓
TRANSLATEPoint D(3, 7) is translated by (+4, −5). Find D′.
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Model Answer
Add the vector to the point: D′ = (3 + 4, 7 + (−5)) = (7, 2)
EXPLAINDoes a triangle with sides 6, 8, and 11 satisfy Pythagoras' theorem?
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Model Answer
Check: 6² + 8² = 36 + 64 = 100. But 11² = 121.
Since 100 ≠ 121, the triangle does NOT satisfy Pythagoras' theorem. It is not a right-angled triangle.
Since 100 ≠ 121, the triangle does NOT satisfy Pythagoras' theorem. It is not a right-angled triangle.
Mathematics · Review
Flashcard Review
Tap each card to reveal the answer. Try to answer from memory first.
State Pythagoras' theorem.
In a right-angled triangle: a² + b² = c², where c is the hypotenuse (longest side, opposite the right angle).
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What is a translation?
A slide — every point moves the same distance in the same direction. Described by a vector (x, y). Shape, size, and orientation are unchanged.
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What is a reflection?
A flip over a line of symmetry. Shape and size stay the same but orientation is reversed.
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What is a rotation?
A turn around a centre point by a given angle. Must state angle, direction, and centre.
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What is a scale factor?
The ratio by which all lengths are multiplied in an enlargement. SF > 1 enlarges; 0 < SF < 1 reduces. Angles stay the same.
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Congruent vs Similar?
Congruent = same shape AND size. Similar = same shape but different size. Translations, reflections, and rotations give congruent images; enlargements give similar images.
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Reflection in the y-axis rule?
(x, y) → (−x, y). The x-coordinate changes sign; the y-coordinate stays the same.
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90° anticlockwise rotation rule?
(x, y) → (−y, x). About the origin.
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180° rotation rule?
(x, y) → (−x, −y). About the origin. Both coordinates change sign.
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How do you find a shorter side using Pythagoras?
Rearrange: a² = c² − b². Subtract the known leg squared from the hypotenuse squared, then square root.
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What is the hypotenuse?
The longest side of a right-angled triangle, always opposite the right angle.
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Common Pythagorean triple: 3, 4, ?
5. Because 3² + 4² = 9 + 16 = 25 = 5². Other common triples: 5-12-13, 8-15-17.
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Which transformation changes size?
Enlargement. The other three (translation, reflection, rotation) keep the size the same.
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How to find distance between two points?
Use Pythagoras: d = √[(x&sub2;−x&sub1;)² + (y&sub2;−y&sub1;)²]. Find horizontal and vertical distances, then apply the theorem.
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Does Pythagoras work for all triangles?
No — only for RIGHT-ANGLED triangles. Always check for a 90° angle before applying it.
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Mathematics · Practice Test
Practice Test — 20 Questions
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