**NCERT Class 9 Maths Lab Manual – Verify that the Ratio of the Areas of a Parallelogram**

**OBJECTIVE**

To verify that the ratio of the areas of a parallelogram and a triangle on the same base and between the same parallels is 2 : 1.

**Materials Required**

- A plywood piece
- Graph paper
- Colour box
- Two wooden strips
- Cutter
- Adhesive
- Geometry box

**Prerequisite Knowledge**

- Area of a triangle.
- Area of a parallelogram.

**Theory**

- For area of triangle refer to Activity 21.
- For area of parallelogram refer to Activity 19.

**Procedure**

- Take a rectangular plywood piece of suitable size and by using adhesive, paste a graph paper on it.
- Take two wooden strips or wooden scale and fix these two horizontally so that they are parallel.
- Fix a pair of points P and Q on the base strip and take a pair of points R and S on the another

strip such that PQ = PS. (see Fig. 22.1) - Take any point T on the second strip and join it to P and Q. (see Fig. 22.1)

- T is any point on RS and PQ is parallel to RS.
- We find that ΔTPQ and parallelogram PQRS lie on the same base PQ and between the same parallels, (see Fig. 22.1)

**Note:**

We may take different triangles TPQ by taking different positions of point T and the two parallel strips, (see Fig. 22.2)

**Demonstration**

- Count the number of squares contained in each of the above ΔTPQ and parallelogram PQRS, keeping half square as ½ and more than half square as 1, leaving those squares which are less than half square.
- We can conclude that the area of the ΔTPQ is half of the area of parallelogram PQRS.

**Observation**

- The number of squares in ΔTPQ = ………….. ,
- The number of squares in parallelogram PQRS = …………. ,

Then, the area of parallelogram PQRS = 2 (area of ΔTPQ).

Hence, area of parallelogram PQRS : area of ΔTPQ = …………. ,

**Result**

We find that the ratio of the area of a parallelogram and the area of a triangle on the same base and between the same parallels is 2 : 1.

**Application**

This activity can be used in

- deriving formula of the area of a triangle.
- solving some problems of mensuration.

**Viva Voce**

**Question 1:**

If a triangle and a parallelogram are on the same base and between the same parallels, then how can we relate the area of triangle and parallelogram?

**Answer:**

Area of the triangle is half the area of parallelogram.

**Question 2:**

If a triangle and a parallelogram are on the same base and having the equal area, then will they have same altitudes?

**Answer:**

No, they will not have same altitudes.

**Question 3:**

If a triangle and a parallelogram are on the same base and between same parallels, then what would be ratio of the area of the triangle to area of parallelogram?

**Answer:**

Required ratio = 1:2

**Question 4:**

Do we obtain a parallelogram and a triangle, whose area are in ratio 2:1?

**Answer:**

Yes, when a parallelogram and a triangle should be on the same base and between same parallels.

**Question 5:**

How can we find the area of a parallelogram with the help of a triangle?

**Answer:**

Area of a parallelogram = 2 x Area of a triangle

( which made by one of the diagonal of parallelogram)

**Question 6:**

How can we find the altitude of a parallelogram?

**Answer:**

Altitude of a parallelogram is the perpendicular drawn from one line to the another parallel line.

**Question 7:**

A triangle and a parallelogram on the same base and between the same parallels, will have same altitudes?

**Answer:**

Yes, they will have same altitudes because distance between two parallel lines remain same at all the points.

**Suggested Activity**

To verify experimentally the relationship between the areas of a parallelogram and a triangle on the same base and between the same parallels by cut out method.

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