A Trapezium and its properties

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Concept # 1. A Trapezium and its properties

Learning objectives

1. A quadrangle with only two opposite sides parallel is called a trapezium, or trapezoid.
2. The parallel sides are called the bases of the trapezium and the other two sides are called the legs or the lateral sides.
3. If the legs are equal in length, then this is an isosceles trapezium.
4. The distance between the bases is called height of trapezium.

Activity No # 1. Properties of a trapezium.

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• Estimated Time :40 minutes.
• Materials/ Resources needed:

Laptop, geogebra file, projector and a pointer.

• Prerequisites/Instructions, if any

1. Knowledge about parallel lines, angles and diagnols.

• Multimedia resources: Laptop.
• Website interactives/ links/ / Geogebra Applets: This file has been taken from www.geogebratube.org

• Process:

1. Show the geogebra file and explain its properties by asking the questions listed below.

• Developmental Questions:

1. What is a trapezium ?
2. How many sides does it have ?
3. What can be said about the pair of sides ?
4. Identify the diagnols. Are they equal ?
5. What would be the sum of 4 angles of the trapezium ? Why so ?
6. What would be the sum of its two allied angles ?
7. Are the two pairs of sides of a trapezium equal ?

• Evaluation:

Are the students able to apply the theorems of parallel lines - transversal to prove the properties.

1. Why are allied angles of the trapezium supplementary ?
2. Are the diagnols equal ?
3. Is trapezium a parallelogram ?

• Question Corner:

1. List the properties of a trapezium.
2. What properties of other quadrilaterals does it share. Compare with other quadrilaterals.

Activity No #

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• Website interactives/ links/ / Geogebra Applets
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• Evaluation
• Question Corner

Concept #2. Measurements in Trapezium

Learning objectives

1. The trapeium contains two parallel sides and two non-parallel sides.
2. The area of trapezium is found by viewing it as a parallelogram.
3. The area of trapezium is 1/2(a+b)h where a and b are its parallel sides and h is the perpendicular distance between them.
4. The perimeter of a trapezium is obtained by sum of its 4 sides.

Notes for teachers

1. The area of most figures can be expressed in terms of its dimensions.
2. The area of most composite figures can be calculated using the area of primary figures.

Activity No # 1.Deriving formula for area of a trapezium

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• Estimated Time: 20 minutes.
• Materials/ Resources needed : Laptop, geogebra file, projector and a pointer.
• Prerequisites/Instructions, if any

1. The students should know a parallelogram and formula to find its area.
2. They should know the trapezium and its properties.

• Multimedia resources: Laptop
• Website interactives/ links/ / Geogebra Applets: This geogebra file has been taken from :www.geogebratbe.org

• Process:

1. The teacher can initially discuss about a trapezium.
2. She can then reiterate that formula for area of certain composite figures can be found by converting them into known simple figures.
3. Here the trapezium is converted into a parallelogram.
4. Area of parallelogram is then deduced.

• Developmental Questions:

1. What is a trapezium ?
2. Name its two parallel sides.
3. What is meant by the height of the trapezium ?
4. After cutting the trapezium exactly in the centre what would be the new height ?
5. What is the length of the new parallelogram formed ?
6. What is the formula to find the area of a parallelogram ?
7. What is the area of this parallelogram formed from a trapezium ?

• Evaluation:

1. Explain the sequence of steps involved in deriving the formula for the area of a trapeium.

• Question Corner:

1. Write down the steps involved in deriving the formula for area of trapezium.

Concept # 3.Construction of Trapezium

Learning objectives

1. Construction of a trapezium accurately for the given measurements.

Activity No #

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• Estimated Time: 40 minutes.
• Materials/ Resources needed: a ruler, pencil, compass, and a blank piece of paper
• Prerequisites/Instructions, if any

1. Knowledge about using a compass and drawing perpendicular lines.

• Multimedia resources
• Website interactives/ links/ / Geogebra Applets

This has been taken from the website ; http://www.k6-geometric-shapes.com/trapezoid.html

• Process:

1. This construction can be made if the height of the trapezium along with the length of the four sides is known.
2. Draw a straight line lightly using your ruler and pencil on your paper. - This is what is called a construction line, and will be the base of the trapezium.
3. Indicate the two end points of the base of the shape with two points, measured by ruler.
   

Note:We know that the top and base line of a trapezium are parallel, and, we know the distance between them (the height).
4. Using compass, construct two lines (lightly) perpendicular to the base. On both of these lines measure the height of the trapezium, and indicate with two points (one on each perpendicular) Connect these two points using a light construction line. The second side of the trapezium will be 'somewhere' on this line.
5. Using ruler, measure the length of the third and fourth sides between these two paralell lines, using the points on the first construction line, as one end of the line segment. Mark with a point.
Note: You will now have four construction lines, intersecting at four vertices.
6. The required trapezium is the shape contained between the four points of intersection of these four lines.
7. Using a heavier line connect the four points to finish your shape.

• Developmental Questions:

1. Name the given parameters of a trapezium.
2. What would be the base ?
3. What are the measures of its 2 parallel sides ?
4. What is the measure of its height ?
5. How would you start the construction process ?
6. what is the method to draw perpendicular lines ?
7. How do you mark the height of the trapezium ?

• Evaluation:

The teacher can check the students constructions to evaluate.

• Question Corner:

1. Can you construct a trapezium when only its area is known ? Discuss.

Activity No #

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Concept # 4. Isosceles trapezium

Learning objectives

1. A trapezium in which non-parallel sides are equal is called as an Isosceles Trapezium.
2. The diagonals of an isosceles trapezium are equal.
3. An isosceles trapezium has one line of reflection symmetry. This line connects the midpoints of the two bases.
4. Both pairs of base angles of an isosceles trapezium are congruent.
5. Pairs of angles in an isosceles trapezium that do not share a base are supplementary.
6. Area of isosceles trapezium is given by$(a+b)/2xh$ , where a and b are the lengths of the parallel sides and h is the distance (height) between the parallel sides.

Activity No # Construct an isosceles trapezium and study its properties

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• Estimated Time: 40 minutes.
• Materials/ Resources needed: Laptop, geogebra file, projector and a pointer.
• Prerequisites/Instructions, if any

1. The students should know the concepts of parallel lines, perpendicular lines and rectangle.
2. They should know basic constructions like parallel lines and perpendicular lines.

• Multimedia resources: Laptop.
• Website interactives/ links/ / Geogebra Applets : This geogebra file has been done by ITfC-Edu-Team

• Process:

1. Recall the figure trapezium and its properties.
2. State that a trapezium with two non- parallel sides equal is an isosceles trapezium.
3. By moving the vertices of the trapezium, you can observe trapeziums of different sizes and shapes.
4. Make sure you note when your trapezium turns into a rectangle.
5. Observe the symmetry of an isosceles trapezium.
6. Study its properties.
7. Drag the vertices of the trapezium and observe your angle measures.
8. Make a conjecture about the base angles of an isosceles trapezium. (Both of the parallel sides are considered bases, so a trapezium has two pairs of base angles.)

• Developmental Questions:

1. What are parallel lines ?
2. What is a trapezium ?
3. Is trapezium a quadrilateral ?
4. What are the characteristic properties of a trapezium ?
5. What do you notice about the non-parallel sides ?
6. How many interior angles do you see ?
7. What is the sum of 4 angles of any quadrilateral ?
8. What can you conclude about interior angles ?
9. What is special about diagnols in an isosceles trapezium ?

• Evaluation:

1. Are all trapeziums isosceles ?
2. Are all trapeziums quadrilaterals too ?
3. Can rectangle be considered as an isosceles trapezium ?

• Question Corner:

1. State the properties of isosceles trapezium ?

Activity No #

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• Estimated Time
• Materials/ Resources needed
• Prerequisites/Instructions, if any
• Multimedia resources
• Website interactives/ links/ / Geogebra Applets
• Process/ Developmental Questions
• Evaluation
• Question Corner

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