Constructions 1.pptx

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Constructions 1

Transformations and constructions

Lesson 6 of 8

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Warm-up

Transformations and constructions

Lesson 6 of 8

Answer each one, then check.

1

What do the angles in a triangle add up to?

2

What is a perpendicular line?

3

What does bisect mean?

4

What is the longest side of a right-angled triangle called?

5

What is the midpoint of a line 8 cm long?

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Warm-up - Answers

Transformations and constructions

Lesson 6 of 8

1

What do the angles in a triangle add up to?

180°

2

What is a perpendicular line?

One that meets another at 90°

3

What does bisect mean?

Cut exactly in half

4

What is the longest side of a right-angled triangle called?

The hypotenuse

5

What is the midpoint of a line 8 cm long?

The point 4 cm from each end

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Learning Objectives

Transformations and constructions

Lesson 6 of 8

1

Construct a triangle given three sides (SSS).

2

Construct a triangle given two sides and the angle between them (SAS).

3

Construct the perpendicular bisector of a line segment.

4

Explain why a triangle can or cannot be constructed.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

The Rules of Construction

Transformations and constructions

Lesson 6 of 8

Use only a pencil, a ruler for straight lines, and compasses.

Keep the construction arcs

Do not rub them out: they show your method and earn marks.

Sharp pencil

Thin lines make the intersections easy to see.

Keep the compasses fixed

Do not change the radius between two arcs that must be equal.

Label the lengths

Write the length beside each side, and mark any angles.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Constructing a Triangle from Three Sides

Transformations and constructions

Lesson 6 of 8

The point where the two arcs cross is the third vertex.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Constructing an SSS Triangle

Transformations and constructions

Lesson 6 of 8

For a triangle with sides 6 cm, 5 cm and 4 cm.

1

Draw the longest side

Measure 6 cm with a ruler and label the ends A and B.

2

Set the compasses to 5 cm

Put the point at A and draw an arc above the line.

3

Set the compasses to 4 cm

Put the point at B and draw a second arc crossing the first.

4

Mark C where the arcs cross

Join C to A and C to B with a ruler.

5

Label and check

Measure to check each side and keep the arcs.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Can the Triangle Be Made?

Transformations and constructions

Lesson 6 of 8

Can a triangle be constructed with sides 3 cm, 4 cm and 8 cm?

1

The two shorter sides must add up to more than the longest

3 + 4 = 7

2

Compare with the longest side

7 < 8

3

The arcs would not meet

The two short sides cannot reach each other

ANSWER

No: 3 + 4 < 8, so the arcs do not cross and no triangle exists.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

An SAS Triangle

Transformations and constructions

Lesson 6 of 8

Construct a triangle with sides of 5 cm and 7 cm and an angle of 40° between them.

1

Draw the 7 cm side

Label the ends A and B

2

Measure 40° at A

Use a protractor and mark the angle

3

Draw a line from A through the mark

Measure 5 cm along it and label C

4

Join B to C

The triangle ABC is complete

ANSWER

Triangle ABC with AB = 7 cm, AC = 5 cm and angle A = 40°.

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The Perpendicular Bisector

Transformations and constructions

Lesson 6 of 8

Every point on the perpendicular bisector is the same distance from A and B.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Constructing the Perpendicular Bisector

Transformations and constructions

Lesson 6 of 8

The same radius from both ends.

1

Open the compasses more than half of AB

Any radius bigger than half will do

2

Draw arcs from A above and below the line

Keep the same radius

3

Draw arcs from B, crossing the first arcs

Do not change the radius

4

Mark the two crossing points

One above and one below AB

5

Join the crossing points with a ruler

This line bisects AB at 90°

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Which Construction?

Transformations and constructions

Lesson 6 of 8

SSS TRIANGLE

• You know all three sides.

• Use two arcs to find the third vertex.

• Always check the two shorter sides add to more than the longest.

SAS TRIANGLE

• You know two sides and the angle between them.

• Use a protractor for the angle.

• The third side is found by joining the ends.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Key Terms

Transformations and constructions

Lesson 6 of 8

Construct

Draw accurately using only a ruler and compasses (a protractor when an angle is given).

Compasses

An instrument for drawing arcs and circles.

Arc

Part of the circumference of a circle.

Bisect

To cut exactly in half.

Perpendicular bisector

A line at 90° through the midpoint of a segment.

Vertex

A corner of a shape.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Transformations and constructions

Lesson 6 of 8

YOUR TASK

Construct and Measure

15 minutes

Construct a triangle with sides 7 cm, 6 cm and 5 cm. Measure the three angles with a protractor and add them up. Then construct the perpendicular bisector of the 7 cm side and check it passes through the opposite vertex or not.

1

Construct with compasses.

2

Measure the angles.

3

Check the sum.

WHAT A GOOD ANSWER SHOWS

The angles are about 78°, 57° and 44° (sum 180°). The perpendicular bisector of the 7 cm side does not pass through the opposite vertex because the triangle is scalene; it would only do that for an isosceles triangle with equal sides on either side.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Can I...?

Transformations and constructions

Lesson 6 of 8

Draw a line of exact length.

Construct an SSS triangle.

Construct an SAS triangle.

Use a protractor accurately.

Construct a perpendicular bisector.

Keep construction arcs.

Test whether three lengths make a triangle.

Explain each step of a construction.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Summary & Exam Focus

SSS: draw the longest side, then two arcs.

SAS: draw one side, measure the angle, then measure the second side.

The perpendicular bisector cuts a segment in half at 90°.

Always leave your construction lines.

EXAM FOCUS

Using ruler and compasses only, construct the perpendicular bisector of the line AB. You must show all your construction lines. (2 marks)

Use the same compass setting from both ends, more than half of AB. Do not rub out the arcs.

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