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NCERT Solutions for Class 7 Maths Chapter 10 Practical Geometry Exercise 10.2

Practical Geometry is an important chapter in Class 7 Mathematics that teaches students how to construct various geometrical shapes using different tools. Exercise 10.2 specifically focuses on constructing triangles when given certain elements of triangles like sides and angles.

Overview of Exercise 10.2

In Exercise 10.2, students learn to construct triangles under different conditions:

  • Constructing triangles when three sides are given (SSS construction)
  • Constructing triangles when two sides and the included angle are given (SAS construction)
  • Constructing triangles when two angles and the included side are given (ASA construction)
  • Constructing right-angled triangles when the hypotenuse and one side are given
  • Constructing triangles when sum/difference of two sides and the base angles are given

Detailed Solutions

Question 1

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

Solution:

We need to construct a triangle when three sides are given (SSS criteria).

  1. Draw a line segment XY = 4.5 cm using a ruler.
  2. With X as center and radius = 6 cm, draw an arc using a compass.
  3. With Y as center and radius = 5 cm, draw another arc using the compass.
  4. Both arcs intersect at a point Z.
  5. Join XZ and YZ using a ruler.

The required triangle XYZ with sides XY = 4.5 cm, YZ = 5 cm and ZX = 6 cm is constructed.

Question 2

Construct an equilateral triangle of side 5.5 cm.

Solution:

An equilateral triangle has all sides equal. We need to construct it with each side = 5.5 cm.

  1. Draw a line segment AB = 5.5 cm using a ruler.
  2. With A as center and radius = 5.5 cm, draw an arc.
  3. With B as center and radius = 5.5 cm, draw another arc.
  4. The arcs intersect at point C.
  5. Join AC and BC.

ABC is the required equilateral triangle with all sides = 5.5 cm.

Question 3

Draw PQR with PQ = 4 cm, QR = 3.5 cm and PR = 4 cm. What type of triangle is this?

Solution:

  1. Draw a line segment PQ = 4 cm.
  2. With P as center and radius = 4 cm, draw an arc.
  3. With Q as center and radius = 3.5 cm, draw another arc.
  4. The arcs intersect at point R.
  5. Join PR and QR.

The constructed triangle has two sides of equal length (PQ = PR = 4 cm). Therefore, this is an isosceles triangle.

Question 4

Construct ABC such that AB = 2.5 cm, BC = 6 cm and AC = 6.5 cm. Measure B.

Solution:

  1. Draw a line segment BC = 6 cm.
  2. With B as center and radius = 2.5 cm, draw an arc.
  3. With C as center and radius = 6.5 cm, draw another arc.
  4. The arcs intersect at point A.
  5. Join AB and AC.
  6. Measure angle B using a protractor.

Upon measuring, we find that B = 90. This triangle is a right-angled triangle with B = 90.

Note: We can verify this by checking if the square of the longest side equals the sum of squares of the other two sides: 6.5 = 42.25, 6 + 2.5 = 36 + 6.25 = 42.25. Since they are equal, by Pythagoras theorem, the triangle is right-angled at B.

Question 5

Construct ABC, given mA = 60, mB = 30 and AB = 5.8 cm.

Solution:

We need to construct a triangle when two angles and the included side are given (ASA criteria).

  1. Draw a line segment AB = 5.8 cm.
  2. At point A, construct an angle of 60 using a protractor.
  3. At point B, construct an angle of 30 on the same side of AB using the protractor.
  4. Extend the rays from A and B to intersect at point C.
  5. Join AC and BC.

ABC is the required triangle with A = 60, B = 30 and AB = 5.8 cm.

Question 6

Construct PQR if QR = 3cm, Q = 90 and PR - PQ = 1.5 cm.

Solution:

We need to construct a right-angled triangle when one leg, the right angle, and the difference of the hypotenuse and other leg are given.

  1. Draw a line segment QR = 3 cm.
  2. At point Q, construct a right angle (90) using a protractor.
  3. From Q, draw a ray QX in the direction of the angle.
  4. From R, draw an arc with radius = 1.5 cm intersecting the ray QX at point S.
  5. Draw the perpendicular bisector of RS, which will intersect the ray QX at point P.
  6. Join PR.

PQR is the required right triangle with Q = 90, QR = 3 cm and PR - PQ = 1.5 cm.

Question 7

Construct a triangle ABC in which BC = 8 cm, B = 45 and AB - AC = 3.5 cm.

Solution:

We need to construct a triangle when a base, a base angle, and the difference of the other two sides are given.

  1. Draw a line segment BC = 8 cm.
  2. At point B, construct an angle of 45 using a protractor.
  3. From B, draw a ray BX making the 45 angle.
  4. From C, mark a point D on BX such that BD = 3.5 cm.
  5. Join CD.
  6. Construct the perpendicular bisector of CD, which intersects BX at point A.
  7. Join AC.

ABC is the required triangle with BC = 8 cm, B = 45 and AB - AC = 3.5 cm.

Question 8

Construct a triangle XYZ in which Y = 30, Z = 90 and XY + YZ + ZX = 11 cm.

Solution:

We need to construct a triangle when two angles and the sum of all three sides (perimeter) are given.

  1. Draw a line segment PQ = 11 cm (which represents XY + YZ + ZX).
  2. At point P, construct an angle of 30.
  3. At point Q, construct an angle of 90 on the same side of PQ.
  4. Draw the angle bisectors of P and Q, which meet at point Z.
  5. Draw the perpendicular bisectors of PZ and QZ, which meet PQ at points X and Y respectively.
  6. Join XZ, YZ, and XY.

XYZ is the required triangle with Y = 30, Z = 90, and perimeter = 11 cm.

Key Concepts in Exercise 10.2

Through Exercise 10.2, students learn several important concepts in geometry:

  • Triangle Construction Criteria: Understanding the conditions under which a triangle can be uniquely determined (SSS, SAS, ASA conditions).
  • Using Geometrical Tools: Developing proficiency in using a ruler, compass, and protractor accurately.
  • Angle Construction: Creating specific angles accurately using protractors or angle bisectors.
  • Verification of Properties: Checking if constructed triangles have the required properties and measurements.
  • Special Triangles: Recognition and construction of special triangles like equilateral, isosceles, and right-angled triangles.

Tips for Accurate Construction

  • Always use sharp pencils for precise markings.
  • Ensure that the compass is properly tightened to avoid changing the radius during construction.
  • Double-check all measurements before drawing final lines.
  • Practice drawing with a steady hand for smoother curves and straighter lines.
  • Use a protractor carefully to ensure accurate angle measurements.
  • Try to minimize the movement of the compass once the radius is set.

Applications of Practical Geometry

The skills learned in Practical Geometry have numerous real-world applications:

  • Architectural designs and construction plans
  • Engineering drawings and technical illustrations
  • Crafting and art projects requiring precise measurements
  • Navigation and map-making
  • Computer-aided design (CAD) basics
  • Robotics and mechanical design

Common Mistakes to Avoid

  • Not reading the problem carefully and using wrong measurements
  • Incorrect placement of the compass while drawing arcs
  • Failing to maintain the correct angle while using a protractor
  • Not double-checking whether the constructed figure meets all given conditions
  • Rushing through the construction without adequate precision
  • Not verifying the final triangle against the given requirements

Conclusion

NCERT Class 7 Maths Chapter 10 Exercise 10.2 on Practical Geometry provides students with foundational skills in geometrical constructions. By learning to construct triangles using different criteria, students develop spatial reasoning, precision, and problem-solving abilities that are essential in advanced mathematics and various practical applications.

These construction techniques not only aid in academic success but also form the basis for numerous real-world applications in fields such as engineering, architecture, design, and technology. The ability to visualize and construct accurate geometrical figures is a valuable skill that extends beyond the mathematics classroom.

Regular practice of these constructions helps students build confidence in geometry and prepares them for more complex geometrical concepts in higher classes. Students are encouraged to practice these constructions multiple times to develop accuracy and speed in their geometric constructions.

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