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

From Karnataka Open Educational Resources

(→Evaluation at the end of the activity) (Tag: Visual edit) |
(→Process (How to do the activity)) (Tag: Visual edit) |
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=== Materials/ Resources needed === | === Materials/ Resources needed === | ||

− | Digital - Computer, Geogebra application, projector. Geogebra | + | Digital - Computer, Geogebra application, projector. Geogebra files- Mid-point theorem1.ggb and Mid-point theorem2.ggb |

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

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

− | Work | + | Work sheet |

Each group member will construct on her / his notebook, using pencil, scale, protractor, and compass a triangle, with the measures provided | 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. | + | # Students should plot the mid-point of two sides and connect these with a line segment. Show [[:File:Mid-point theorem 1.ggb|Mid-point theorem1.ggb]] step by step. |

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

− | # Prove the theorem | + | # Prove the theorem, using [[:File:Mid-point theorem 2.ggb|Mid-point theorem2.ggb]] |

## In △ ABC, D and E are the midpoints of sides AB and AC respectively. D and E are joined. | ## 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. | ## Given: AD = DB and AE = EC. To Prove: DE ∥∥ BC and DE = 1/2 BC. |

## Revision as of 15:51, 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 files- Mid-point theorem1.ggb and Mid-point theorem2.ggb

Non digital - worksheet and pencil.

### Process (How to do the activity)

Work sheet

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. Show Mid-point theorem1.ggb step by step.
- 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, using Mid-point theorem2.ggb
- 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?