Lesson Plans SET A for Grades 9-10-Lesson 10 Level 9-10 3D Geometry

Lesson 10 Level 9-10 3D Geometry - HW

Pdf Summary

The document provides a set of problems for practicing 3D geometry concepts, specifically aiming at Math Kangaroo level 9-10. Here is a summary of each problem and its solution:<br /><br />1. **Triangle in 3D Space**: Given the vertices of a triangle, the task is to find the measure of angle A using side lengths calculated via the distance formula and then applying the cosine rule. The solution shows the calculation steps, leading to the conclusion that angle A is \(90^\circ\).<br /><br />2. **Spheres and a Plane**: This problem involves finding the vertical distance from a plane to the top of a large sphere resting on three smaller, tangent spheres. By recognizing the geometry involving the spheres' radii and the geometric arrangement (equilateral triangle), the solution uses the Pythagorean Theorem to find the height contributed by geometric arrangements, ultimately totaling \(4 + \frac{2\sqrt{3}}{3}\).<br /><br />3. **Cone in Water**: The task is to find the depth of water over a partially submerged cone. By understanding the volume ratios (only \(\frac{1}{8}\) above water), the height submerged is calculated as \(50\) feet.<br /><br />4. **Cube Cross Section**: A cube is cut through two opposite vertices, creating a rhombus cross-section. Using the Pythagorean theorem, the diagonals of this rhombus are determined, from which the area is calculated as \(\frac{6}{2}\).<br /><br />5. **3D Shapes**: This is more of a construction problem where students are asked to draw a 3D view of a composite object based on various perspectives provided. An example view is indicated, suggesting creativity in interpreting the perspectives.<br /><br />These exercises challenge students to apply various concepts in three-dimensional geometry, including coordinate geometry, spatial reasoning, volume and ratio understanding, and visualization skills.

Keywords

3D geometry

Math Kangaroo

triangle in space

spheres and plane

cone in water

cube cross section

coordinate geometry

spatial reasoning

volume and ratio

visualization skills