Chapter 4

Practical Geometry

 

Introduction

In Class VII, you learned how to draw triangles. To draw a unique triangle, you need three measurements (sides and angles). As three measurements are sufficient to create a triangle, it’s natural to wonder if four measurements would be enough to draw a unique four-sided closed figure, such as a quadrilateral.

NCERT Solutions for Class 8 Maths Chapter 4 Practical Geometry are beneficial to students because they assist them in achieving good scores in exams. NCERT Solutions include extensive step-by-step explanations for all of the questions in Chapter 4 of the Class 8 NCERT Textbook’s Practical Geometry section. Students will be able to overcome all of their worries about practical geometry by comprehending principles employed in NCERT Solutions for Class 8.

Take a pair of equal-length sticks of say 10 cm. Take a second pair of equal-length sticks, this time of say 8 cm. Hinge them together to make a rectangle of length 10 cm and width 8 cm. These four sticks of known measurements can thus form a rectangle. 

Now, simply press along the length of the rectangle. Is the new shape that you have created still a rectangle? No, it has transformed into a parallelogram. Did you make any changes to the stick lengths? No! The side measurements are the same. What happens if you give the newly formed shape another shove in a different direction? You obtain a different parallelogram but with the same measurements. This activity demonstrates that a quadrilateral cannot be determined solely by its measures.

We can make different quadrilaterals using the following measurements:

• When there are four sides and one diagonal

• When there are two diagonals and three sides

• When there are two neighbouring sides and three angles

• When there are three sides and two included angles

• When other unique characteristics are known

Let’s pick each of these formations one by one to understand them better.

Here is an example to demonstrate the formation of a quadrilateral using this construction.

Example 1: PQ = 4 cm, QR = 6 cm, RS = 5 cm, PS = 5.5 cm and PR = 7 cm are the dimensions of a quadrilateral PQRS.

(A rough sketch will assist you in visualising the quadrilateral.) 

Step 1: Draw PQR using the SSS construction condition.

Step 2: Now, find the fourth point, S. Concerning PR, it should be on the side opposite to Q. There are two measurements for this. P is 5.5 cm apart from S. Draw an arc with a radius of 5.5 cm and with P as a centre. (Point S lies somewhere on this arc.)

Step 3: S is 5 cm apart from R. Draw an arc with a radius of 5 cm, with R as a centre (Point S is also on this arc.)

Step 4: S should be positioned on both of the drawn arcs. It is the place where the two arcs meet. Mark S and finish PQRS. 

The required quadrilateral is PQRS.

(i) We have seen that 5 quadrilateral measurements can be used to determine a quadrilateral uniquely. Do you believe any five quadrilateral measurements will be enough?

(ii) Can you draw a parallelogram BATS with BA = 5 cm, AT = 6 cm and AS = 6.5 cm? Why?

(iii) Can you draw a rhombus ZEAL with ZE = 3.5 cm and EL = 5 cm on the diagonal? Why?

When you are given four sides and a diagonal, first draw a triangle and then find the fourth point. A method identical to the previous case is used here.

Example 2: Construct a quadrilateral ABCD with BC = 4.5 cm, AD = 5.5 cm, CD = 5 cm, diagonal AC = 5.5 cm and diagonal BD = 7 cm. 

Solution: Draw a rough sketch of quadrilateral ABCD. It shows that it is possible to draw ACD first.

Step 1: Create ACD using the SSS method. (At this point, locate B at a distance of 4.5 cm from C and 7 cm from D.)

Step 2: Draw an arc with a radius of 7 cm and with D as a centre. (B lies at some point along this arc.)

Step 3: Draw an arc with a radius of 4.5 cm and with C as a centre (B is also on this arc).

Step 4: B is the point of intersection of the two arcs because it lies on both. Complete ABCD by marking B. 

The required quadrilateral is ABCD.

To complete the quadrilateral, we begin by creating a triangle and then searching for the fourth point.

Example 3: Make a quadrilateral MIST with MI = 3.5 cm, IS = 6.5 cm and T = 7.5 cm.

M is 75°, I is 105° and S is 120°.

Step 1: How do you figure out where the points are? What would you use as a base and what will be the initial step?

Step 2: Set ISY to 120° at S.

Step 3: Set IMZ to 75° at M. (Which location will SY and MZ meet?) Make a T in that spot. The required quadrilateral MIST is obtained.

When drawing a basic sketch of this type, pay close attention to the included angles.

Example 4: Make a quadrilateral ABCD with AB = 4 cm, BC = 5 cm, CD = 6.5 cm, angle B = 105° and angle C = 80°.

Solution: As a general rule, begin with a preliminary sketch to locate the four points of the required quadrilateral.

Step 1: Begin by measuring BC = 5 cm on B. Draw a 105° angle along BX. On this, find A at a distance of 4 cm. Points B, C and A are now available.

Step 2: The fourth point D is on CY, which is at an angle of 80° to BC. Set angle BCY = 80° on BC.

Step 3: D is at a distance of 6.5 cm from CY. Draw a 6.5-cm long arc with C at a centre. It intersects CY at D.

Step 4: Complete the quadrilateral. 

The required quadrilateral ABCD is thus formed.

In the aforementioned cases, five measurements were used to draw a quadrilateral. Is there any quadrilateral that can be drawn using fewer measurements? The following examples demonstrate some of these unique situations.

Example 5: Make a square with a side of 4.5 cm.

Solution: At first glance, it appears that only one measurement has been provided. Since the required quadrilateral is a square, this information is sufficient. 

You know that each angle of a square is a right angle. Thus, you can draw ABC using SAS conditions. Then, D will be easy to find. Now, draw the square using the measurements provided.

Example 6: Can a rhombus ABCD with AC = 6 cm and BD = 7 cm be constructed? Justify your response.

Solution: Only two measurements (diagonals) are given. This information is sufficient to construct a rhombus. 

The diagonals of a rhombus are perpendicular bisectors of each other.

Step 1: Draw AC = 7 cm, and then mark its perpendicular bisector.

Step 2: Let them meet at 0. Sections of 3 cm should be cut off on each side of the bisector. These are points B and D.

Step 3: On CY, D is at a distance of 6.5 cm. Draw a 6.5-cm long arc with C as a centre.

It intersects CY at D.

1. In the preceding example, you drew BC first. What other possible starting points can you think of?

2. So far, you have used five measurements to draw quadrilaterals. Is it possible to draw a quadrilateral using alternative sets of five measurements?

The problems given below may assist you in solving the question.

(i) In quadrilateral ABCD, AB = 5 cm, BC = 5.5 cm, CD = 4 cm, AD = 6 cm and angle B = 80°.

(ii) In quadrilateral PQRS, PQ = 4.5 cm, angle P = 70°, angle Q = 100°, angle R = 80° and angle S = 110°.

Try to come up with a few examples of your own to determine whether the data is sufficient to create a quadrilateral.

Practical geometry is a field of mathematics that deals with the evaluation of various shapes, sizes and other attributes of geometric figures. It also entails the formation of various geometrical designs to find precise and simple problem-solving methods.

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