NCERT Solutions for Class 7 Maths Chapter 10 Practical Geometry

Introduction to Practical Geometry

Practical Geometry is an important branch of mathematics that deals with the construction of various geometrical figures. NCERT Solutions for Class 7 Maths Chapter 10 Practical Geometry helps students understand how to construct different geometrical shapes using various tools like ruler, compass, protractor, and divider.

The chapter focuses on the construction of:

• Line segments
• Circles
• Triangles
• Quadrilaterals

Learning Practical Geometry helps develop spatial understanding and logical reasoning, which are essential skills for higher mathematics. This chapter introduces students to step-by-step construction methods and the properties of different geometric shapes.

Key Concepts in Practical Geometry

1. Construction of a Line Segment

A line segment can be constructed if its length is given. The steps involve using a ruler to measure and draw the line segment of the given length.

2. Construction of a Circle

A circle can be constructed if its radius is given. A circle is the set of all points in a plane that are at a given distance from a fixed point called the center.

3. Construction of Triangles

A triangle can be uniquely constructed when three of its six elements (three sides and three angles) are known, with at least one being a side. Different criteria for constructing triangles include:

• S-S-S Criterion (Three sides given)
• S-A-S Criterion (Two sides and included angle given)
• A-S-A Criterion (Two angles and included side given)
• R-H-S Criterion (Right triangle with hypotenuse and one side given)

4. Construction of Quadrilaterals

A quadrilateral can be constructed uniquely when five of its elements (four sides and one diagonal, or four sides and one angle, etc.) are known. Different methods of constructing quadrilaterals are discussed in this chapter.

Exercise 10.1 Solutions

Question 1: Draw a line segment of length 7.3 cm using a ruler.

Ans. Steps of construction:

1. Draw a line.
2. Mark a point A on the line.
3. Using a ruler, measure 7.3 cm from point A.
4. Mark point B at the measured distance.
5. Join A and B to get the line segment AB of length 7.3 cm.

Question 2: Construct a line segment of length 5.6 cm using ruler and compasses.

Ans. Steps of construction:

1. Draw a line.
2. Mark a point A on the line.
3. Open the compass to a length of 5.6 cm.
4. Place the compass pointer at point A and draw an arc cutting the line.
5. Mark the intersection point as B.
6. AB is the required line segment of length 5.6 cm.

Question 3: Given AB = 3 cm and CD = 5 cm, construct a line segment XY such that the length of XY is equal to the difference between the lengths of AB and CD.

Ans. First, we find the difference between the lengths of AB and CD:

CD = 5 cm, AB = 3 cm

Difference = CD - AB = 5 cm - 3 cm = 2 cm

Now, to construct a line segment XY of length 2 cm:

1. Draw a line.
2. Mark a point X on the line.
3. Using a ruler, measure 2 cm from point X.
4. Mark point Y at the measured distance.
5. Join X and Y to get the line segment XY of length 2 cm.

Exercise 10.2 Solutions

Question 1: Construct XYZ in which XY = 4.5 cm, YZ = 5 cm and ZX = 6 cm.

Ans. Steps of construction using SSS criterion:

1. Draw a line segment ZX = 6 cm.
2. With Z as center and radius 5 cm, draw an arc.
3. With X as center and radius 4.5 cm, draw another arc to intersect the previous arc at Y.
4. Join ZY and XY.
5. XYZ is the required triangle.

Question 2: Construct an isosceles triangle in which the lengths of each of its equal sides is 6.5 cm and the angle between them is 110.

Ans. Steps of construction:

1. Draw a line segment AB = 6.5 cm.
2. At point A, draw an angle of 110 using a protractor.
3. On the ray making the angle, mark point C such that AC = 6.5 cm.
4. Join B and C.
5. ABC is the required isosceles triangle with AB = AC = 6.5 cm and A = 110.

Question 3: Construct PQR with PQ = 4 cm, QR = 3.5 cm and PR = 4 cm. What type of triangle is this?

Ans. Steps of construction:

1. Draw a line segment QR = 3.5 cm.
2. With Q as center and radius 4 cm, draw an arc.
3. With R as center and radius 4 cm, draw another arc to intersect the previous arc at P.
4. Join QP and RP.
5. PQR is the required triangle.

This is an isosceles triangle because two sides (PQ and PR) are equal in length.

Exercise 10.3 Solutions

Question 1: Construct DEF such that DE = 5 cm, DF = 3 cm and EDF = 90.

Ans. Steps of construction:

1. Draw a line segment DE = 5 cm.
2. At point D, construct a right angle and draw a ray DX.
3. With D as center and radius 3 cm, draw an arc to cut the ray DX at point F.
4. Join E and F.
5. DEF is the required right-angled triangle.

Question 2: Construct an isosceles right-angled triangle ABC, where ACB = 90 and AC = 6 cm.

Ans. Steps of construction:

1. Draw a line segment AC = 6 cm.
2. At point C, construct a right angle and draw a ray CX.
3. With C as center and radius 6 cm, draw an arc to cut the ray CX at point B.
4. Join A and B.
5. ABC is the required isosceles right-angled triangle.

Question 3: Construct ABC, given ABC = 60, BC = 5 cm and AB = 4.5 cm.

Ans. Steps of construction using SAS criterion:

1. Draw a line segment BC = 5 cm.
2. At point B, construct an angle of 60 and draw a ray BX.
3. With B as center and radius 4.5 cm, draw an arc to cut the ray BX at point A.
4. Join A and C.
5. ABC is the required triangle.

Tips and Tricks for Solving Practical Geometry Problems

• Always keep your geometry tools clean and in good condition. A rusty compass or blunt pencil can affect your constructions.
• Mark measurements clearly and label all points properly to avoid confusion.
• Before starting the construction, read the problem carefully and identify what is given and what needs to be constructed.
• Remember the different criteria for triangle construction (SSS, SAS, ASA, RHS) and apply the appropriate one based on the given data.
• Use light pencil markings initially so that you can erase if needed. Use darker lines only for the final construction.
• Practice constructions regularly to develop a good sense of precision and accuracy.
• Always verify your construction using the properties of the geometric figure you've created.
• When constructing angles, ensure your protractor is properly aligned with the baseline for accurate measurements.