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

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=== Objectives ===
 
=== 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
+
# To guide and facilitate student exploration of the properties of triangles, specifically – 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
+
# To introduce theorems associated with triangles
 +
# To demonstrate the steps of logical proof and the processes
 +
# Understand properties of triangles
  
 
=== Estimated Time ===
 
=== Estimated Time ===
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=== Prerequisites/Instructions, prior preparations, if any ===
 
=== Prerequisites/Instructions, prior preparations, if any ===
−
Prior knowledge of point, lines, angles, parallel lines, triangles, and quadrilaterals / parallelograms
+
An understanding of the basic elements of geometry.  Students are familiar with angles and parallel lines, and have been introduced to the concepts of parallelogram
  
 
=== Materials/ Resources needed ===
 
=== Materials/ Resources needed ===
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Work sheet
 
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. You can make chits that have triangle measures.  You can even repeat them.  For a group give the following. One acute.  One obtuse. One equilateral. One isosceles. One right.  Provide only measures, for e.g.
 +
# Draw  a triangle with following measures  - BC =  6 cms, AC = 7 cms, AB = 8 cms (BC as the base)
 +
# Draw  a triangle with following measures  BC =  8 cms, angle B = 110, AB = 6 cms (BC as the base)
 +
# Draw  a triangle with following measures  BC =  10 cms, angle B = 60, angle C = 60 (BC as the base)
 +
# Draw  a triangle with following measures  BC =  12 cms, angle B = 90, AC = 13 cms (BC as the base)
 +
# Draw  a triangle with following measures  BC =  8 cms, angle B = 40, angle C = 40 (BC as the base)
 +
# Draw  a triangle with following measures  BC =  6 cms, angle B = 110, AB = 5 cms (BC as the base)
 +
# Draw  a triangle with following measures  BC =  8 cms, angle B = 60, angle C = 40 (BC as the base)
  
−
# 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.
+
Each group gets this set and each member picks one chit and makes a sketch
−
## They should measure the length of this segment and length of the third side and check if there is any relationship
+
# Students should plot the mid-point of two sides (Segment AB, and AC, as D and E) and connect these with a line segment (Join these mid-points D and E.  Label it Segment DE). Show [[:File:Mid-point theorem 1.ggb|Mid-point theorem1.ggb]] step by step.
−
## 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 length of this segment DE and length of the third side BC and check if there is any relationship. They should measure the angles formed at the two vertices B and C connecting the third side, with the two angles formed on the two mid-points, D and E. For this each student will make three tables, each table  will have the following columns -
−
# Question them if there is any relationship between the two segment lengths and the measures of the two pairs of angles.  
+
## Table for vertex A + Sides from the vertex,  Third side, Midpoint segment name, midpoint segment length, comparison with the side length.
−
## Ask them why these relationships are true across different constructions.  
+
## Angle formed at midpoint 1, angle at vertex B, the angle at vertex C (students should name the angle and measure). Each student completes it for each vertex.
−
# Prove the theorem, using [[:File:Mid-point theorem 2.ggb|Mid-point theorem2.ggb]]
+
{| class="wikitable"
 +
|'''Vertex'''
 +
|'''Base'''
 +
|'''Length of base'''
 +
|'''Length of side 1'''
 +
|'''Length of side 2'''
 +
|'''Length of segment connecting mid-points'''
 +
|'''Relationship between 5 and 6'''
 +
|'''Angle at vertex 2'''
 +
|'''Angle at vertex 3'''
 +
|'''Angle at D'''
 +
|'''Angle at E'''
 +
|'''Relationship between 8 and 10, 9 and 11'''
 +
|-
 +
|1
 +
|2
 +
|3
 +
|4
 +
|5
 +
|6
 +
|7
 +
|8
 +
|9
 +
|10
 +
|11
 +
|12
 +
|-
 +
|A
 +
|BC
 +
|
 +
|AC=
 +
|AB=
 +
|DE=
 +
|
 +
|
 +
|
 +
|
 +
|
 +
|
 +
|-
 +
|B
 +
|AC
 +
|
 +
|BC=
 +
|AB=
 +
|DE=
 +
|
 +
|
 +
|
 +
|
 +
|
 +
|
 +
|-
 +
|C
 +
|AB
 +
|
 +
|BC=
 +
|AC=
 +
|DE=
 +
|
 +
|
 +
|
 +
|
 +
|
 +
|}
 +
# Ask them to see if there is any relationship between the two segment lengths and the measures of the two pairs of angles.  
 +
# Ask them if these relationships are true across different constructions. Can they look at the observations and state it as a rule or a statement
 +
# Tell students that we will use properties of parallel lines and parallelogram for this proof (to help them get into the thinking about the process of deduction, required in proving theorems).  
 +
# Prove the theorem, using [[:File:Mid-point theorem 2.ggb|Mid-point theorem2.ggb]] (students can follow the demonstration through Geogebra and also write the steps in their book, relating to the triangle that they have drawn)
 
## 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.  
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# By the properties of parallelogram, we have DF ∥∥ BC DF = BC DE ∥∥ BC.  
 
# By the properties of parallelogram, we have DF ∥∥ BC DF = BC DE ∥∥ BC.  
 
## DE = 1/2BC  (DE = EF by construction)
 
## 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]]

Latest revision as of 13:51, 7 November 2019

Objectives

  1. To guide and facilitate student exploration of the properties of triangles, specifically – 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.
  2. To introduce theorems associated with triangles
  3. To demonstrate the steps of logical proof and the processes
  4. Understand properties of triangles

Estimated Time

One period

Prerequisites/Instructions, prior preparations, if any

An understanding of the basic elements of geometry.  Students are familiar with angles and parallel lines, and have been introduced to the concepts of parallelogram

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. You can make chits that have triangle measures.  You can even repeat them.  For a group give the following. One acute.  One obtuse. One equilateral. One isosceles. One right.  Provide only measures, for e.g.

  1. Draw a triangle with following measures - BC = 6 cms, AC = 7 cms, AB = 8 cms (BC as the base)
  2. Draw a triangle with following measures BC = 8 cms, angle B = 110, AB = 6 cms (BC as the base)
  3. Draw a triangle with following measures BC = 10 cms, angle B = 60, angle C = 60 (BC as the base)
  4. Draw a triangle with following measures BC = 12 cms, angle B = 90, AC = 13 cms (BC as the base)
  5. Draw a triangle with following measures BC = 8 cms, angle B = 40, angle C = 40 (BC as the base)
  6. Draw a triangle with following measures BC = 6 cms, angle B = 110, AB = 5 cms (BC as the base)
  7. Draw a triangle with following measures BC = 8 cms, angle B = 60, angle C = 40 (BC as the base)

Each group gets this set and each member picks one chit and makes a sketch

  1. Students should plot the mid-point of two sides (Segment AB, and AC, as D and E) and connect these with a line segment (Join these mid-points D and E.  Label it Segment DE). Show Mid-point theorem1.ggb step by step.
  2. They should measure the length of this segment DE and length of the third side BC and check if there is any relationship. They should measure the angles formed at the two vertices B and C connecting the third side, with the two angles formed on the two mid-points, D and E. For this each student will make three tables, each table  will have the following columns -
    1. Table for vertex A + Sides from the vertex,  Third side, Midpoint segment name, midpoint segment length, comparison with the side length.
    2. Angle formed at midpoint 1, angle at vertex B, the angle at vertex C (students should name the angle and measure). Each student completes it for each vertex.
Vertex Base Length of base Length of side 1 Length of side 2 Length of segment connecting mid-points Relationship between 5 and 6 Angle at vertex 2 Angle at vertex 3 Angle at D Angle at E Relationship between 8 and 10, 9 and 11
1 2 3 4 5 6 7 8 9 10 11 12
A BC AC= AB= DE=
B AC BC= AB= DE=
C AB BC= AC= DE=
  1. Ask them to see if there is any relationship between the two segment lengths and the measures of the two pairs of angles.
  2. Ask them if these relationships are true across different constructions. Can they look at the observations and state it as a rule or a statement
  3. Tell students that we will use properties of parallel lines and parallelogram for this proof (to help them get into the thinking about the process of deduction, required in proving theorems).
  4. Prove the theorem, using Mid-point theorem2.ggb (students can follow the demonstration through Geogebra and also write the steps in their book, relating to the triangle that they have drawn)
    1. In △ ABC, D and E are the midpoints of sides AB and AC respectively.  D and E are joined.
    2. Given: AD = DB and AE = EC. To Prove: DE ∥∥ BC and DE = 1/2 BC.
    3. Construction: Extend line segment DE to F such that DE = EF. Draw segment CF.
    4. Proof: In △ ADE and △ CFE AE = EC  (given)
    5. ∠AED = ∠CEF (vertically opposite angles)
    6. DE = EF   (construction)
    7. Hence △ ADE ≅ △ CFE by SAS congruence rule.
      1. Therefore, ∠ADE = ∠CFE  (by CPCT) and ∠DAE = ∠FCE (by CPCT) AD = CF (by CPCT).
  5. ∠ADE and ∠CFE are alternate interior angles,
  6. (AB and CF are 2 lines intersected by transversal DF).
  7. ∠DAE and ∠FCE are alternate interior angles,
  8. (AB and CF are 2 lines intersected by transversal AC).
    1. Therefore, AB ∥∥ CF. So - BD ∥∥ CF.
  9. BD = CF (since AD = BD and it is proved above that AD = CF).
    1. Thus, BDFC is a parallelogram.
  10. By the properties of parallelogram, we have DF ∥∥ BC DF = BC DE ∥∥ BC.
    1. DE = 1/2BC  (DE = EF by construction)

Evaluation at the end of the activity

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