Geometry & Statistics
Transformations, Pythagorean Theorem, and Volume of 3D Shapes
Grade 8
- ✓Describe and perform translations, reflections, rotations, and dilations on two-dimensional figures.
- ✓Understand and apply the concepts of congruence and similarity using transformations.
- ✓Apply the Pythagorean Theorem to determine unknown side lengths in right triangles and to find the distance between two points in a coordinate system.
- ✓Solve real-world and mathematical problems involving the volumes of cones, cylinders, and spheres.
Key concepts
A transformation is a change in the position, size, or orientation of a figure. There are four main types of transformations:
A transformation that slides a figure from one position to another without turning it. The orientation and size of the figure remain the same.
A transformation that flips a figure over a line, called the line of reflection. The reflected figure is a mirror image of the original, with the same size and shape.
A transformation that turns a figure around a fixed point, called the center of rotation, by a specific angle. The size and shape of the figure remain the same.
A transformation that changes the size of a figure by a scale factor, either enlarging or shrinking it, but maintains its shape. The center of dilation is the fixed point from which the figure is enlarged or shrunk.
Two figures are congruent if one can be obtained from the other by a sequence of rigid transformations (translations, reflections, or rotations). Congruent figures have the same size and the same shape.
Two figures are similar if one can be obtained from the other by a sequence of rigid transformations and a dilation. Similar figures have the same shape but not necessarily the same size. Their corresponding angles are equal, and their corresponding side lengths are proportional.
In a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides (legs).
If the square of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right-angled triangle.
The amount of space occupied by a cylinder. It is calculated by multiplying the area of its circular base by its height.
The amount of space occupied by a cone. It is one-third the volume of a cylinder with the same base radius and height.
The amount of space occupied by a sphere.
Key facts to remember
- 1Rigid transformations (translations, reflections, rotations) preserve the size and shape of a figure, resulting in congruent figures.
- 2Dilations change the size of a figure but preserve its shape, resulting in similar figures.
- 3The Pythagorean Theorem (a² + b² = c²) applies only to right-angled triangles, where 'c' is always the hypotenuse.
- 4The converse of the Pythagorean Theorem can be used to determine if a triangle is a right triangle.
- 5Volume formulas: Cylinder V = πr²h; Cone V = (1/3)πr²h; Sphere V = (4/3)πr³.
- 6The volume of a cone is exactly one-third the volume of a cylinder with the same radius and height.
Worked examples
Example 1
Triangle ABC has vertices A(1,2), B(3,2), C(2,4).\na) Reflect triangle ABC across the y-axis to create triangle A'B'C'. Are triangle ABC and triangle A'B'C' congruent or similar?\nb) Dilate triangle ABC by a scale factor of 2 with the origin as the center of dilation to create triangle A''B''C''. Are triangle ABC and triangle A''B''C'' congruent or similar?
Answer
a) Congruent. b) Similar.
Example 2
A ladder is leaning against a wall. The base of the ladder is 5 feet from the wall, and the ladder reaches 12 feet up the wall. How long is the ladder?
Answer
The ladder is 13 feet long.
Always ensure you identify the hypotenuse (c) correctly as the side opposite the right angle.
Example 3
Calculate the volume of a cylinder with a radius of 4 cm and a height of 10 cm. Then, calculate the volume of a cone with the same radius and height, and the volume of a sphere with a radius of 4 cm. Use π ≈ 3.14.
Answer
Volume of cylinder ≈ 502.4 cm³. Volume of cone ≈ 167.47 cm³. Volume of sphere ≈ 267.95 cm³.
Remember to use the correct units (cubic centimeters) for volume and to round as specified.
Common mistakes
- ✗Confusing congruence with similarity, especially when a dilation is involved.
- ✗Incorrectly identifying the hypotenuse in the Pythagorean Theorem, or applying it to non-right triangles.
- ✗Forgetting to square the sides or take the square root at the end when using the Pythagorean Theorem.
- ✗Mixing up the volume formulas, particularly forgetting the (1/3) for cones or (4/3) for spheres.
- ✗Using diameter instead of radius (or vice-versa) in volume calculations without adjusting.
Exam tips
- ★Always draw a diagram for geometry problems, especially for transformations and the Pythagorean Theorem, to visualize the problem.
- ★Memorize the key formulas for the Pythagorean Theorem and the volumes of cylinders, cones, and spheres.
- ★When using the Pythagorean Theorem, clearly label the legs (a, b) and the hypotenuse (c) to avoid errors.
- ★Pay close attention to units in volume problems; remember that volume is measured in cubic units (e.g., cm³, ft³).
- ★Show all your steps in calculations, especially for multi-step problems, to earn partial credit even if the final answer is incorrect.
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