## 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). 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.
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 ?
Area of triangle = $$\frac { 1 }{ 2 }$$ x base x height

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

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

Question 5.
Is mid-point theorem applicable in any type of triangle ?
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 ?
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 ?
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?
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
(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

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

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