Geometry and Measurement · Shape and Space · Lesson 10
Perpendiculars and Bisectors in the Environment
Right angles, midpoints and equal halves are not just ideas from your geometry set. They hold up buildings, shape roofs, fix electricity poles and mark out a football pitch. In this lesson you will find perpendicular lines, perpendicular bisectors and angle bisectors in the world around you, and practise estimating angles.
By the end you will be able to
- ✔Identify perpendicular line segments in buildings, tools and objects around you.
- ✔Identify and describe perpendicular bisectors and angle bisectors in structures such as roofs, tanks and boxes.
- ✔Estimate the size of angles in artefacts and structures using benchmark angles.
- ✔Explain why these ideas matter to builders, carpenters and surveyors.
Key words
- Perpendicular
- Meeting at a right angle (90°).
- Perpendicular bisector
- A line that cuts a segment in half at 90°.
- Angle bisector
- A line that cuts an angle into two equal angles.
- Plumb line
- A string with a weight, hanging straight down; it shows the vertical.
- Benchmark angle
- A well-known angle (such as 90° or 45°) used to estimate others.
Curriculum reference
Content standard: B7.3.1.2 Demonstrate how to construct a perpendicular to a line from a given point, bisect a line, bisect angles, and construct angles of the following sizes: 30°, 45°, 60°, 75° and 90°.
Indicator: Describe examples of perpendicular line segments, perpendicular bisectors and angle bisectors in the environment.
Starter
Geometry on the building site
At a building site in Tamale, the mason checks the wall with a plumb line, the carpenter fixes a brace across the corner of the window frame, and the roofing team centres the ridge post on the middle of the beam. Every one of these jobs uses a geometry idea you have constructed with compasses. Can you spot them? Look around your own classroom after this lesson.
1. Perpendicular lines around you
Geometry is not only in your exercise book. Look around your classroom. The edge of the wall meets the floor at 90°. The leg of your desk stands at 90° to the floor. A window frame, a door, a signboard and a football pitch corner all show perpendicular line segments. Two line segments are perpendicular when they meet at a right angle.
| Object | Perpendicular segments | Why it matters |
|---|---|---|
| Classroom wall and floor | The edge of the wall and the floor | The wall stands straight and does not lean |
| Electricity pole (ECG) | The pole and the ground | A leaning pole may fall in a storm |
| Door frame | The sides and the top bar | The door fits and closes properly |
| Football pitch | The goal line and the touchline | Gives a rectangular field |
| Plumb line (builder's string with a weight) | The string and the ground | Shows a true vertical, perpendicular to the ground |
How do builders check for a right angle? They use a set square, a builder's square, a plumb line (a weight hanging on a string), or the 3-4-5 rope method. All of them depend on the idea of a 90° angle that you constructed in the previous lessons.
2. Perpendicular bisectors in structures and artefacts
A perpendicular bisector cuts a segment into two equal parts at 90°. Many designs depend on it. The central post of a symmetric roof, a fold in a piece of cloth that makes the two edges meet, and the vertical line down the middle of a rectangular water tank are all perpendicular bisectors of the base.
In the roof below, the triangle ABC is symmetrical (isosceles). The post CM goes from the top vertex to the midpoint of the base. It is perpendicular to the base, so it is the perpendicular bisector of AB.
▶ Watch what the ridge post CM does: it halves AB, it meets AB at 90°, and it halves the angle at C.
- A pole in the middle of a rectangular market shed that stands at the midpoint of the front beam is on its perpendicular bisector.
- A folded sheet of paper: if you fold a line segment on top of itself so that the two ends meet, the crease is the perpendicular bisector.
- The centre line of a road is the perpendicular bisector of a straight cross-section of the road if the road is symmetrical.
▶ The line PQ is the perpendicular bisector of AB: each point on it is equally far from A and B.
3. Angle bisectors in the environment
An angle bisector cuts an angle into two equal angles. You find them wherever something is symmetrical. The diagonal brace in the corner of a square window frame, the crease along the diagonal of a folded square handkerchief, and the line that divides a slice of cake into two equal slices are all angle bisectors.
▶ A corner of 90° is cut by the bisector into two 45° angles, like the diagonal brace of a window frame.
| Structure or artefact | Angle | The bisector shows up as |
|---|---|---|
| Corner of a square window frame | 90° | The diagonal brace, which makes two 45° angles |
| Gable end of a roof | e.g. 80° at the top | The ridge post, which makes two 40° angles |
| Pair of scissors | Variable | The line along which the blades are symmetric |
| Folded handkerchief | 90° corner | The crease along the diagonal |
| Water tank square base | 90° corner | The diagonal from corner to corner |
The diagonal of a square is both: it bisects the 90° angle at the corner (45° each) and it is also a line of symmetry. The same is true for a rhombus. Not every shape is like this: in a rectangle that is not a square, the diagonal does not bisect the corner angle.
4. Estimating angles in everyday objects
The curriculum also asks you to estimate the size of angles in tools, artefacts and structures. To estimate well, use benchmark angles that you know: 90° (a corner), 45° (half a corner), 60° and 30°, and 180° (a straight line). Then compare.
| Benchmark | Angle | Everyday example |
|---|---|---|
| Corner of a book | 90° | Page corner, door corner |
| Half a corner | 45° | Diagonal brace, cut corner of a handkerchief |
| Straight line | 180° | Fully open book, a table top |
| Clock hands at 3 o'clock | 90° | Quarter of a turn |
| Each hour on a clock face | 30° | 360° ÷ 12 |
| Leaning ladder | about 75° with the ground | A safe angle for a ladder |
| Roof slope | about 15° to 30° | Roofing sheet fixed on a rafter |
Estimate first, then measure. Estimate the angle between the hands of a clock at 5 o'clock. Each hour is 30°, and 5 hours is 5 × 30° = 150°. Then check by measuring with a protractor, and see how close your estimate was. With practice you will estimate to within 10°.
Worked examples
Try each one yourself first, then open the solution and compare every step.
Example 1
Identify the perpendicular segments
Show the solution
- Step 1Look for corners that make a right angle.
- Step 2Examples: the edge of the wall and the floor; the leg of a desk and the floor; the top edge and the side edge of the blackboard.
- Step 3All of these make 90° corners.
Answer: Wall and floor, desk leg and floor, blackboard edges
Example 2
The roof post
Show the solution
- Step 1M is the midpoint of AB, so CM cuts AB into two equal parts.
- Step 2CM is perpendicular to AB, so the angle at M is 90°.
- Step 3A line that does both is the perpendicular bisector.
Answer: CM is the perpendicular bisector of AB
Example 3
The window brace
▶ The diagonal brace is an angle bisector of the corner. What are the two parts?
Show the solution
- Step 1The corner of a square is 90°.
- Step 2The diagonal bisects the corner angle.
- Step 390° ÷ 2 = 45°.
Answer: 45° with each side
Example 4
Estimate a clock angle
Show the solution
- Step 1On a clock face there are 12 hours in 360°, so each hour is 360° ÷ 12 = 30°.
- Step 2At 5 o'clock the hands are 5 hours apart.
- Step 35 × 30° = 150°.
Answer: 150°
Example 5
Ladder against a wall
Show the solution
- Step 1Wall and ground are perpendicular, so the angle between them is 90°.
- Step 2The ladder splits this 90° angle into 75° and the rest.
- Step 3Angle with wall = 90° − 75° = 15°.
Answer: 15°
Summary
- Perpendicular line segments meet at 90°: wall and floor, pole and ground, door frame corners.
- A perpendicular bisector cuts a segment in half at 90°: the centre post of a symmetric roof, a paper fold line.
- An angle bisector cuts an angle into two equal angles: the diagonal brace in a square corner (45° each).
- To estimate an angle, compare it to benchmark angles of 90°, 45°, 30° and 180°.
- A clock face gives 30° per hour, so 3 o'clock is 90° and 5 o'clock is 150°.
Practice: 15 questions
5 easy, 5 medium, 5 hard. Each one is marked straight away, with a full worked solution.
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