The Pythagorean Theorem, Transformations and Volume: Grade 8 Study Guide
Geometry in Grade 8 links shapes, algebra and the coordinate plane.
In a right triangle, the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the legs: a² + b² = c². The 3-4-5 triangle works: 9 + 16 = 25. The converse is also true: if a² + b² = c², the triangle is a right triangle. The theorem finds distances in the coordinate plane and unknown sides in problems such as a ladder against a wall.
Transformations move figures. A translation slides, a reflection flips, a rotation turns and a dilation enlarges or shrinks using a scale factor. The first three produce congruent figures; a dilation produces a similar figure.
Using the theorem
A 13 ft ladder rests against a wall with its base 5 ft away. The height is √(13² − 5²) = √144 = 12 ft. The distance between (1, 2) and (7, 10) is √(6² + 8²) = √100 = 10.
Right angle, hypotenuse c.
Fill in known sides.
Take a square root.
c is the longest side.
Transformations on the plane
Reflect (3, 2) in the y-axis: (−3, 2). Translate (3, 2) by 4 right and 1 down: (7, 1). Rotate (3, 2) 90° counterclockwise about the origin: (−2, 3). Dilate (3, 2) by a scale factor of 2 from the origin: (6, 4). In similar triangles, corresponding angles are equal and sides are proportional.
| Transformation | Rule on (x, y) | Result for (3, 2) |
|---|---|---|
| Reflect in x-axis | (x, −y) | (3, −2) |
| Reflect in y-axis | (−x, y) | (−3, 2) |
| Rotate 90° counterclockwise | (−y, x) | (−2, 3) |
| Dilate by 2 | (2x, 2y) | (6, 4) |
Volume of solids
Cylinder: V = πr²h. Cone: V = ⅓πr²h. Sphere: V = (4/3)πr³. A cone has one-third the volume of a cylinder with the same base and height. A sphere of radius 3 has V = (4/3)π(27) = 36π ≈ 113 cubic units. Use these for tanks, cans and balls.
Worked examples and common mistakes
Example: A can has r = 4 cm and h = 10 cm: V = π × 16 × 10 = 160π ≈ 503 cm³. Common mistakes: using a² + b² = c² with a non-right triangle, forgetting to take the square root and using diameter for radius. Sketch the triangle first.
Check your understanding: model answers
1. 15. 2. (−4, −5). 3. ⅓π(9)(7) = 21π.
✨ Key Takeaways
- a² + b² = c² for right triangles.
- Transformations: translate, reflect, rotate, dilate.
- Similar figures have equal angles and proportional sides.
- Cone = ⅓ cylinder.
💬 Questions for Discussion & Reflection
- A right triangle has legs 9 and 12. Find the hypotenuse.
- Reflect (−4, 5) in the x-axis.
- Find the volume of a cone with r = 3 and h = 7 (leave π).