Topic 5 Lines and Planes in Three Dimensions

Key Words

dimensions

plane

projection

By the end of this topic, you should be able to:

1. apply Pythagoras’ theorem in three dimensions to calculate the distance between two points.
2. find the angle between a line and a plane.
3. find the angle between two planes.

Introduction

In real life, many of the things that you see, such as a desktop computer, a chalkboard, a piece of paper, a TV screen, a window, a wall, and a door, are examples of planes. However, when you look at the ends of edges of those objects or surfaces, an idea of lines comes in. In Senior Two, you explored areas and lengths of 2-dimensional figures.

In Topic 11, you learnt about nets, areas, and volumes of solids. Building on that experience, this topic will help you to understand and, thereafter, apply the knowledge acquired to solve problems.

Activity 5.0(a) (Work in groups) (a) Study the following shapes:

(b) Identify the faces of each shape by shading.

(c) What number of faces does each shape above have?

(d) Draw a table showing the formulae of the volume and surface area of each shape above.

(e) Share your findings with other groups.

Activity 5.0(b) (Work in groups)

(a) Explore your environment to identify more solids in your location.

(b) For each solid identified in (a), count and record the number of vertices, edges and faces.

(c) Determine the relationship among the numbers of vertices, edges and faces. (d) Present your work to the class.

5.1 Applying Pythagoras’ Theorem in 3-D to Calculate the Distance between Two Points

As discussed earlier, Pythagoras’ theorem applies to right-angled triangles. Therefore, if any three points connect to create a right-angled triangle, then, the distances between the three points are related by Pythagoras’ theorem.

Activity 5.1(a) (Work in groups)

Suggested materials:

a manilla paper

a cutter

geometrical construction instruments

sellotape

Instructions:

(a) Construct a net of a square-based pyramid.

(b) Compute the height of the constructed pyramid.

(c) Display your work to the class.

Activity 5.1(b) (Work in groups)

Suggested Materials:

a manilla paper

a cutter

geometrical construction instruments

Instructions:

(a) Construct a cube. sellotape

(b) Compute the length of the diagonal in a face and that across the cube. Confirm the results of your computations by measuring those dimensions on the cube.

(c) What is your observation?

(d) Compute the volume of your model.

(e) How many square-based pyramids can be made from your model?

(f) Present your findings to the rest of the class.

5.2 Finding the Angle between a Line and a Plane Activity 5.2

(a) (Work in groups) Suggested materials: straws/sticks Instructions: a cutter glue or sellotape (a) Use the materials provided to you to construct a cuboid of your choice.

(b) Show the possible diagonals on the cuboid.

(c) Name the angles between the diagonals and any plane on the cuboid.

(d) Share your findings with other groups.

Activity 5.2(b) (Work in groups)

Suggested Materials:

straws/sticks

a cutter

glue or sellotape

Instructions:

(a) Make a square-based pyramid. Strengthen the base and the triangular faces using diagonals.

(b) Identify the angle between:

(i) each slant edge and the base.

(ii) each slant height and the base.

(iii) a pair of slant heights.

(c) Calculate the angle between the perpendicular height and any slant edge.

5.3 Finding the Angle between Two Planes

Two planes meet, if they do so in a line. Do you think that three planes meet in a line too?

Activity 5.3 (Work in groups)

Suggested Materials

a manilla paper

a cutter

geometrical construction instruments

sellotape / paper glue

Instructions:

(a) Construct a net for a square-based pyramid whose perpendicular height is half the length of the square.

(b) Identify the angle between:

(i) any triangular face and the base.

(ii) any two adjacent triangular faces.

(c) How many of these pyramids can fill up a cube?

(d) Compute the volume of the pyramid whose net is constructed in (a) (e) Share with other groups.

5) Arrange eight congruent circles, each with Radius 2 cm, inside a rectangle. Find the perimeter of the rectangle.

6)(a) Roll a piece of paper into a cone shape, so that the tip touches the bottom of a cylinder of your choice.

(b) Tape the cone shape along the sides and trim it to form a cone with the same height as the cylinder.

(c) Fill the cone to the top with sand or sugar, or any other substance, and empty the contents into the cylinder. Repeat this as many times as needed to completely fill the cylinder.

(d) What is the relationship between the volume of the cone and that of the cylinder?

7) Study the figure below and answer the questions that follow:

Topic Summary

In this topic, you have learnt that:

1) Pythagoras’ theorem can be applied even to 3-dimensional figures, by creating right-angled triangles from them.

2) to find the angle between a line and a plane, you locate the projection of the line along that plane first and, thereafter, measure the angle between them.

3) to find the angle between two planes, draw a line through one of the planes from its mid-point to intersect with another line from the other plane. Then, measure the angle between the lines.