# Difference between revisions of "Quadrilaterals-Activity-Mid-point theorem"

From Karnataka Open Educational Resources

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=== Objectives === | === Objectives === | ||

− | Demonstrate that the line joining the mid-points of any two sides of a triangle is parallel to the third side and equals to half the third side | + | # Understand properties of triangles – a segment connecting mid-points of two sides of a triangle will be parallel to the third side and its length will be half of the third side |

+ | # Demonstrate that the line joining the mid-points of any two sides of a triangle is parallel to the third side and equals to half the third side | ||

=== Estimated Time === | === Estimated Time === | ||

Line 6: | Line 7: | ||

=== Prerequisites/Instructions, prior preparations, if any === | === Prerequisites/Instructions, prior preparations, if any === | ||

+ | Prior knowledge of point, lines, angles, parallel lines, triangles, and quadrilaterals / parallelograms | ||

=== Materials/ Resources needed === | === Materials/ Resources needed === | ||

− | Geogebra file | + | Digital - Computer, Geogebra application, projector. Geogebra file - Mid-point theorem.ggb |

+ | |||

+ | Non digital -worksheet and pencil. | ||

=== Process (How to do the activity) === | === Process (How to do the activity) === | ||

+ | Work shee - t | ||

+ | |||

+ | Each group member will construct on her / his notebook, using pencil, scale, protractor, and compass a triangle, with the measures provided | ||

+ | |||

+ | # Students should plot the mid-point of two sides and connect these with a line segment. | ||

+ | ## They should measure the length of this segment and length of the third side and check if there is any relationship | ||

+ | ## They should measure the angles formed at the two vertices connecting the third side, with the two angles formed on the two mid-points | ||

+ | # Question them if there is any relationship between the two segment lengths and the measures of the two pairs of angles. | ||

+ | ## Ask them why these relationships are true across different constructions. | ||

+ | # Prove the theorem | ||

+ | ## In △ ABC, D and E are the midpoints of sides AB and AC respectively. D and E are joined. | ||

+ | ## Given: AD = DB and AE = EC. To Prove: DE ∥∥ BC and DE = 1/2 BC. | ||

+ | ## Construction: Extend line segment DE to F such that DE = EF. Draw segment CF. | ||

+ | ## Proof: In △ ADE and △ CFE AE = EC (given) | ||

+ | ## ∠AED = ∠CEF (vertically opposite angles) | ||

+ | ## DE = EF (construction) | ||

+ | ## Hence △ ADE ≅ △ CFE by SAS congruence rule. | ||

+ | ### Therefore, ∠ADE = ∠CFE (by CPCT) and ∠DAE = ∠FCE (by CPCT) AD = CF (by CPCT). | ||

+ | # ∠ADE and ∠CFE are alternate interior angles, | ||

+ | # (AB and CF are 2 lines intersected by transversal DF). | ||

+ | # ∠DAE and ∠FCE are alternate interior angles, | ||

+ | # (AB and CF are 2 lines intersected by transversal AC). | ||

+ | ## Therefore, AB ∥∥ CF. So - BD ∥∥ CF. | ||

+ | # BD = CF (since AD = BD and it is proved above that AD = CF). | ||

+ | ## Thus, BDFC is a parallelogram. | ||

+ | # By the properties of parallelogram, we have DF ∥∥ BC DF = BC DE ∥∥ BC. | ||

+ | ## DE = 1/2BC (DE = EF by construction) | ||

=== Evaluation at the end of the activity === | === Evaluation at the end of the activity === | ||

− | + | # Would this theorem apply for right angled and obtuse-angled triangles? | |

[[Category:Quadrilaterals]] | [[Category:Quadrilaterals]] | ||

[[Category:Class 9]] | [[Category:Class 9]] |

## Revision as of 10:10, 7 November 2019

## Contents

### Objectives

- Understand properties of triangles – a segment connecting mid-points of two sides of a triangle will be parallel to the third side and its length will be half of the third side
- Demonstrate that the line joining the mid-points of any two sides of a triangle is parallel to the third side and equals to half the third side

### Estimated Time

One period

### Prerequisites/Instructions, prior preparations, if any

Prior knowledge of point, lines, angles, parallel lines, triangles, and quadrilaterals / parallelograms

### Materials/ Resources needed

Digital - Computer, Geogebra application, projector. Geogebra file - Mid-point theorem.ggb

Non digital -worksheet and pencil.

### Process (How to do the activity)

Work shee - t

Each group member will construct on her / his notebook, using pencil, scale, protractor, and compass a triangle, with the measures provided

- Students should plot the mid-point of two sides and connect these with a line segment.
- They should measure the length of this segment and length of the third side and check if there is any relationship
- They should measure the angles formed at the two vertices connecting the third side, with the two angles formed on the two mid-points

- Question them if there is any relationship between the two segment lengths and the measures of the two pairs of angles.
- Ask them why these relationships are true across different constructions.

- Prove the theorem
- In △ ABC, D and E are the midpoints of sides AB and AC respectively. D and E are joined.
- Given: AD = DB and AE = EC. To Prove: DE ∥∥ BC and DE = 1/2 BC.
- Construction: Extend line segment DE to F such that DE = EF. Draw segment CF.
- Proof: In △ ADE and △ CFE AE = EC (given)
- ∠AED = ∠CEF (vertically opposite angles)
- DE = EF (construction)
- Hence △ ADE ≅ △ CFE by SAS congruence rule.
- Therefore, ∠ADE = ∠CFE (by CPCT) and ∠DAE = ∠FCE (by CPCT) AD = CF (by CPCT).

- ∠ADE and ∠CFE are alternate interior angles,
- (AB and CF are 2 lines intersected by transversal DF).
- ∠DAE and ∠FCE are alternate interior angles,
- (AB and CF are 2 lines intersected by transversal AC).
- Therefore, AB ∥∥ CF. So - BD ∥∥ CF.

- BD = CF (since AD = BD and it is proved above that AD = CF).
- Thus, BDFC is a parallelogram.

- By the properties of parallelogram, we have DF ∥∥ BC DF = BC DE ∥∥ BC.
- DE = 1/2BC (DE = EF by construction)

### Evaluation at the end of the activity

- Would this theorem apply for right angled and obtuse-angled triangles?