NCERT Solutions Class VII Maths Chapter 10 Practical Geometry Exercise 10.1
Introduction to Practical Geometry
Practical Geometry is a fascinating branch of mathematics that deals with constructing various geometric figures using only basic tools like a ruler and compass. In Class VII, students learn to construct different types of triangles based on given measurements. This chapter helps develop spatial reasoning and precise drawing skills that are essential in various fields like engineering, architecture, and design.
Chapter 10 of the NCERT Mathematics textbook for Class VII focuses on Practical Geometry, specifically the construction of triangles. Exercise 10.1 introduces students to the step-by-step process of constructing triangles when different sets of measurements are provided.
About Exercise 10.1
Exercise 10.1 of Chapter 10 Practical Geometry focuses on constructing different types of triangles when various measurements are given, such as sides, angles, or a combination of both. The problems require students to apply their knowledge of triangle properties and construction techniques to create accurate geometric figures.
Let's explore each question in Exercise 10.1 and provide detailed step-by-step solutions.
Question 1: Construction of a Triangle When Three Sides are Given (SSS Criterion)
Problem Statement:
Construct a triangle ABC, given AB = 5 cm, BC = 6 cm, and CA = 7 cm.
Steps of Construction:
Explanation:
In this problem, we used the SSS (Side-Side-Side) criterion for triangle construction. When all three sides of a triangle are given, we can construct a unique triangle. The key steps involve drawing one side first and then using arcs to locate the position of the third vertex where the other two sides meet.
This construction demonstrates that if three sides of a triangle are given, then there exists exactly one triangle with these measurements (assuming the triangle inequality is satisfied).
Question 2: Construction of a Triangle When Two Sides and the Included Angle are Given (SAS Criterion)
Problem Statement:
Construct a triangle PQR, given PQ = 4 cm, QR = 5 cm, and PQR = 60.
Steps of Construction:
Explanation:
This problem uses the SAS (Side-Angle-Side) criterion for triangle construction. When two sides and the included angle (the angle between the two sides) are given, we can construct a unique triangle. The construction process involves drawing one side first, then constructing the given angle at one endpoint, and finally marking the length of the other side on the ray of the angle.
The SAS criterion is one of the ways to prove triangle congruence, and it allows us to construct a triangle because the position of all three vertices can be determined uniquely from the given information.
Question 3: Construction of a Triangle When Two Sides and an Angle are Given (SSA Criterion)
Problem Statement:
Construct a triangle XYZ, given XY = 5 cm, XZ = 6 cm, and XYZ = 45.
Steps of Construction:
Explanation:
This problem demonstrates the SSA (Side-Side-Angle) criterion for triangle construction. When two sides and a non-included angle (an angle not between the two sides) are given, there can be:
- No triangle possible (if the given side is too short to reach the opposite side)
- Exactly one triangle possible (if the given side just touches the opposite side)
- Two different triangles possible (if the given side can intersect the opposite side in two different positions)
Question 4: Construction of a Triangle When One Side and Two Angles are Given (ASA Criterion)
Problem Statement:
Construct a triangle LMN, given LM = 6 cm, L = 45, and M = 60.
Steps of Construction:
Explanation:
This problem uses the ASA (Angle-Side-Angle) criterion for triangle construction. When one side and the two adjacent angles are given, we can construct a unique triangle. The construction process involves drawing the given side first, then constructing the two given angles at its endpoints. The intersection of these two rays determines the third vertex of the triangle.
The ASA criterion is particularly useful because it allows us to find the third angle (since the sum of angles in a triangle is 180) and construct the triangle without needing any additional information about the lengths of the other sides.
Question 5: Construction of a Right-Angled Triangle When Hypotenuse and One Side are Given (RHS Criterion)
Problem Statement:
Construct a right-angled triangle ABC, right-angled at B, given hypotenuse AC = 7 cm and side BC = 5 cm.
Steps of Construction:
Explanation:
This problem deals with constructing a right-angled triangle when the hypotenuse and one side are given. Since we know the triangle is right-angled at B, we first draw the given side BC. Then we construct a 90 angle at B and use the length of the hypotenuse AC to locate point A.
This construction demonstrates the Pythagorean theorem, where in a right triangle, the square of the hypotenuse equals the sum of squares of the other two sides. The RHS (Right angle-Hypotenuse-Side) criterion is one of the ways to prove congruence for right triangles and allows us to construct a unique right triangle when the hypotenuse and one side are given.
Key Concepts and Formulas
Triangle Construction Criteria:
1. SSS Criterion: When three sides are given, a unique triangle can be constructed.
2. SAS Criterion: When two sides and the included angle are given, a unique triangle can be constructed.
3. ASA Criterion: When one side and two adjacent angles are given, a unique triangle can be constructed.
4. SSA Criterion: When two sides and a non-included angle are given, 0, 1, or 2 triangles may be possible depending on the specific measurements.
5. RHS Criterion: When the hypotenuse and one side of a right triangle are given, a unique right triangle can be constructed.
Important Triangle Properties:
1. The sum of interior angles in a triangle is always 180.
2. Triangle Inequality: The sum of any two sides of a triangle is greater than the third side.
3. Pythagorean theorem: In a right triangle, the square of the hypotenuse equals the sum of squares of the other two sides (a + b = c).
4. Exterior angle theorem: An exterior angle of a triangle equals the sum of the two opposite interior angles.
Tips and Common Mistakes to Avoid
Helpful Tips:
- Always use a sharp pencil for precise construction.
- Double-check measurements before drawing lines.
- Practice constructing arcs with a compass to achieve smooth curves.
- Light construction lines can be drawn first before finalizing the figure.
- Ensure the compass is properly adjusted to maintain the correct radius during construction.
- Mark measurements clearly and label all vertices and sides correctly.
- Keep your work area clean and organized for better accuracy.
Common Mistakes to Avoid:
- Incorrectly transferring measurements from the scale to the paper.
- Not properly aligning the compass, leading to inaccurate arcs.
- Mixing up the given measurements while constructing.
- Not maintaining the angle correctly while using a protractor.
- Forgetting to label the vertices and sides of the constructed triangle.
- Drawing arcs too lightly or not completing them fully, making intersection points unclear.
- Inconsistent units of measurement for different sides of the triangle.
Conclusion
Practical Geometry is an essential skill that helps students understand the properties of geometric figures through direct experience. Exercise 10.1 has provided practice in constructing different types of triangles using specific criteria like SSS, SAS, ASA, SSA, and RHS.
These constructions not only reinforce theoretical knowledge but also develop precision, patience, and attention to detailskills that are valuable in many areas beyond mathematics. Regular practice with these constructions will improve your geometric intuition and problem-solving abilities.
By mastering the techniques in this exercise, students will be better prepared for more complex geometric constructions in higher classes. The ability to construct accurate geometric figures is fundamental to fields like engineering, architecture, and design, making Practical Geometry a truly practical branch of mathematics.
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