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Mathematics Class 9 - Triangles Notes

Comprehensive study notes for Class 9 - Triangles olympiad preparation

Triangles

In this chapter, you will learn about triangles, their types, properties, and important theorems. By the end of this chapter, you will be able to identify different triangles, use their properties, and solve geometry problems involving triangles.

Key Concepts

  • Triangle: A polygon with three sides and three angles.
  • Vertices: The corners of a triangle.
  • Sides: The straight lines that form the triangle.
  • Angles: The space between two sides at a vertex.

Types of Triangles

  • Based on Sides:
    • Equilateral Triangle: All sides and angles are equal.
    • Isosceles Triangle: Two sides and two angles are equal.
    • Scalene Triangle: All sides and angles are different.
  • Based on Angles:
    • Acute Triangle: All angles are less than 90°.
    • Right Triangle: One angle is 90°.
    • Obtuse Triangle: One angle is more than 90°.

Properties of Triangles

  • The sum of the angles in a triangle is always 180°.
  • The length of any side is less than the sum of the other two sides.
  • The area of a triangle can be found using different formulas.

Important Theorems

  • Pythagoras Theorem: In a right triangle, (Hypotenuse)2 = (Base)2 + (Height)2
  • Congruence of Triangles: Two triangles are congruent if their sides and angles are equal (criteria: SSS, SAS, ASA, RHS).
  • Similarity of Triangles: Two triangles are similar if their corresponding angles are equal and sides are in the same ratio.

Area of a Triangle

  • Area = ½ × Base × Height
  • Heron's Formula: Area = √[s(s-a)(s-b)(s-c)], where s = (a+b+c)/2 and a, b, c are the sides.

Practice Questions

  1. Classify the triangle with sides 5 cm, 5 cm, and 8 cm.
  2. Find the area of a triangle with base 12 cm and height 9 cm.
  3. What is the sum of the angles in any triangle?
  4. State the Pythagoras Theorem.
  5. If two triangles have equal sides and angles, what are they called?

Challenge Yourself

  • Draw a right triangle and label its base, height, and hypotenuse.
  • Use Heron's formula to find the area of a triangle with sides 7 cm, 8 cm, and 9 cm.

Did You Know?

  • Triangles are the strongest shape used in building bridges and towers!
  • The triangle is the only polygon with the minimum number of sides.

Glossary

  • Vertex: A corner of a triangle.
  • Congruent: Exactly equal in shape and size.
  • Similar: Same shape but different size.
  • Hypotenuse: The longest side in a right triangle.

Answers to Practice Questions

  1. Isosceles triangle (two sides equal).
  2. Area = ½ × 12 × 9 = 54 cm²
  3. 180°
  4. In a right triangle, (Hypotenuse)2 = (Base)2 + (Height)2
  5. Congruent triangles.

Practice problems on triangles to master geometry and solve real-life challenges!

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