**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**

- Concept of angles, triangles and mid-points.
**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**

- Draw ∆ABC on the yellow glazed paper of any measurement and paste it on white sheet.
- 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.

- Draw horizontal line DE. Similarly find mid-point of side BC and name it F as shown in fig. (ii).

- Trace the ∆ABC on tracing paper and cut ∆ABC along line DE as shown in fig.(iii).

- 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).

- ∆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**

- (i)
- (i)
- (i)
- (ii)
- (i)
- (iii)
- (i)
- (ii)
- (i)
- (i)

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