CBSE Class 9 Maths Lab Manual – Mid-point Theorem

CBSE Class 9 Maths Lab Manual – Mid-point Theorem

Objective
To verify that in a triangle, the line joining the mid-points of any two sides is parallel to the third side and half of it by paper folding and pasting.

Prerequisite Knowledge

1. Concept of angles, triangles and mid-points.
2. Concept of corresponding angles: If a transversal cuts two straight lines such that their corresponding angles are equal, then the lines are parallel.

Materials Required
Glazed papers, a pair of scissors, pencil, eraser, gluestick, white sheet.

Procedure

1. Draw ∆ABC on the yellow glazed paper of any measurement and paste it on white sheet.
2. Find mid-points of the two sides (say AB and AC) of a triangle by paper folding. We obtain D and E as mid-points of AB and AC respectively in 1st triangle.
   
3. Draw horizontal line DE. Similarly find mid-point of side BC and name it F as shown in fig. (ii).
   
4. Trace the ∆ABC on tracing paper and cut ∆ABC along line DE as shown in fig.(iii).
   
5. Paste this cut out of triangle ADE [fig. (iii) ] on ∆ABC of fig. (ii) such that AE coincides with EC and ED lies on CB and point D coincides with F as shown in fig. (iv).
   
6. ∆ADE completely covers ∆EFC.

Observation
We observe that ∆ADE exacdy overlaps ∆EFC.
∴ ∠1 = ∠2 (corresponding angles)
AC is any transversal line intersecting the lines DE and BC.
∴ DE || BC.
By paper folding we observe that, in fig (iv) F, the mid point of BC coincides with D.
∴ DE = FC (As DE superimposes on FC)
or DE = FC = \(\frac { BC }{ 2 }\)

Result
Hence, it is verified that the line joining the mid-points of two sides of a triangle is parallel to third side and half of it.

Learning Outcome
Line segment joining the mid-points of any two sides of a triangle is parallel to the third side and is equal to half of it. This is true for all types of triangles like acute-angled triangle, obtuse-angled triangle and right-angled triangle.

Activity Time
Students can verify this theorem in different triangles, e.g., obtuse-angled triangle, right-angled triangle, equilateral triangles, scalene triangles.

Viva Voce

Question 1.
State the mid-point theorem.
Answer:
The line drawn through the mid-point of one side of a triangle and parallel to another side of the triangle, bisects the third side of the triangle.

Question 2.
What is the area of a triangle ?
Answer:
Area of triangle = \(\frac { 1 }{ 2 }\) x base x height

Question 3.
Name the different triangles on the basis of its sides.
Answer:
Equilateral triangle, scalene triangle, isosceles triangle.

Question 4.
Name the different triangles on the basis of its angles.
Answer:
Acute angled triangle, obtuse angled triangle and right angled triangle.

Question 5.
Is mid-point theorem applicable in any type of triangle ?
Answer:
Yes.

Question 6.
In a triangle, the line drawn through the mid-point of one side is parallel to another side, what is the ratio of parallel line to the third side ?
Answer:
1:2.

Question 7.
In a triangle, the line drawn through the mid-points of two sides, then what will be the relation between the line and the third side ?
Answer:
Line will be parallel to the third side.

Question 8.
In a ∆ABC, D, E, F are the mid-points of the sides BC, CA and AB respectively, and ∠BAC = 70°, what is the value of ∠EDF?
Answer:
70°

Multiple Choice Questions

Question 1.
In a ∆ABC, P is the mid-point of AB and Q is mid-point of AC and PQ = 4 cm, what will be the length of BC?
(i) 8 cm
(ii) 16 cm
(iii) 2 cm
(iv) none of these

Question 2.
In a right-triangle PQR, A is the mid-point of PQ and B is the mid-point of QR with AB = 5 cm, what will be the length of the hypotenuse PR:
(i) 10 cm
(ii) 25 cm
(iii) 125 cm
(iv) none of these

Question 3.
What is the length of PB, if in a triangle PQR, if A is the mid-point of PR and AB || QR, and PQ = 6 cm ?
(i) 3 cm
(ii) 12 cm
(iii) 18 cm
(iv) none of these

Question 4.
What is the length of AB, if M is the mid-point of AC and LM || BC, in a ∆ABC ?
(i) 2BM
(ii) 2BL
(iii) \(\frac { 1 }{ 2 }\) AB
(iv) none of these

Question 5.
What will be the length of AC, if L is the mid-point of AB and LM || BC, in ∆ABC and AM = 4 cm ?
(i) 8 cm
(ii) 2 cm
(iii) 9 cm
(iv) none of these

Question 6.
In a right-triangle, mid-points of corresponding sides are joined, the resulting triangle will be:
(i) an acute angled triangle
(ii) an obtuse angled triangle
(iii) a right-angled triangle
(iv) none of these

Question 7.
L, M, N are the mid-points of corresponding sides AB, BC, CA in ∆ABC, the figure so obtained BLMN will be:
(i) parallelogram
(ii) trapezium
(iii) quadrilateral
(iv) none of these

Question 8.
In a right-angled triangle PQR, right angled at ∠Q, A, B and C are mid-points of corresponding sides PQ, QR and PR. The figure so obtained ACBQ is a
(i) parallelogram
(ii) rectanlge
(iii) trapezium
(iv) none of these

Question 9.
In a ∆ABC, P, Q and R are the mid-points of corresponding sides AB, BC and CA. Join P, Q and R. Now ar (∆PQR) will be :
(i) \(\frac { 1 }{ 4 }\) ar(∆ABC)
(ii) \(\frac { 1 }{ 2 }\) ar(∆ABC)
(iii) 4 ar(∆ABC)
(iv) none of these

Question 10.
In a ∆PQR, if B and C are the mid-points of sides PR and QR respectively, then BC || PQ and:
(i) BC = \(\frac { 1 }{ 2 }\) PQ
(ii) BC = 2PQ
(iii) BC = \(\frac { 1 }{ 4 }\) PQ
(iv) none of these

Answers

1. (i)
2. (i)
3. (i)
4. (ii)
5. (i)
6. (iii)
7. (i)
8. (ii)
9. (i)
10. (i)

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