Class 7 Notes - The Triangle and its Properties

Download PDF Worksheets
Start a Practice Test

The Triangle and Its Properties

A triangle is a closed figure with three sides, three angles, and three vertices. In this chapter, we will explore the different types of triangles, their properties, and important concepts like medians, altitudes, and the sum of angles in a triangle.

1. Types of Triangles

  • Based on Sides:
    • Equilateral Triangle: All three sides are equal and all angles are 60°.
    • Isosceles Triangle: Two sides are equal and the angles opposite to them are equal.
    • Scalene Triangle: All sides and all angles are different.
  • Based on Angles:
    • Acute-angled Triangle: All angles are less than 90°.
    • Right-angled Triangle: One angle is exactly 90°.
    • Obtuse-angled Triangle: One angle is more than 90°.

2. Properties of a Triangle

  • Angle Sum Property: The sum of the three angles of a triangle is always 180°.
  • Exterior Angle Property: An exterior angle of a triangle is equal to the sum of the two opposite interior angles.
  • Triangle Inequality Property: The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

3. Medians and Altitudes

  • Median: A line segment joining a vertex to the midpoint of the opposite side. Every triangle has 3 medians, and they all meet at a point called the centroid.
  • Altitude: A perpendicular segment from a vertex to the opposite side (or its extension). Every triangle has 3 altitudes, and they meet at a point called the orthocenter.

4. Other Important Terms

  • Centroid: The point where the three medians of a triangle meet. It divides each median in the ratio 2:1.
  • Orthocenter: The point where the three altitudes of a triangle meet.
  • Incenter: The point where the angle bisectors of a triangle meet. It is the center of the circle inscribed in the triangle.
  • Circumcenter: The point where the perpendicular bisectors of the sides meet. It is the center of the circle passing through all three vertices.

5. Example Problems

  1. Find the value of x: In triangle ABC, angle A = 50°, angle B = 60°, find angle C.
    Solution: Angle C = 180° - (50° + 60°) = 70°
  2. Exterior Angle: In triangle PQR, if the exterior angle at Q is 120° and the interior opposite angle at P is 50°, find angle R.
    Solution: Exterior angle = angle P + angle R ⇒ 120° = 50° + angle R ⇒ angle R = 70°
  3. Triangle Inequality: Can a triangle have sides 3 cm, 4 cm, and 8 cm?
    Solution: 3 + 4 = 7 < 8, so these sides cannot form a triangle.

6. Practice Exercises

  1. Classify the following triangles based on their sides: (a) 5 cm, 5 cm, 5 cm (b) 6 cm, 6 cm, 8 cm (c) 7 cm, 8 cm, 9 cm
  2. Find the missing angle: In triangle XYZ, angle X = 80°, angle Y = 60°, angle Z = ?
  3. Draw a triangle and mark its medians, centroid, and altitudes.
  4. State whether a triangle with sides 2 cm, 3 cm, and 6 cm is possible. Why or why not?
  5. In a triangle, one angle is 90°, and another is 45°. What is the third angle?

7. Summary

  • Triangles can be classified by sides and angles.
  • The sum of the angles in a triangle is always 180°.
  • Medians, altitudes, centroid, orthocenter, incenter, and circumcenter are important concepts.
  • The triangle inequality property helps determine if three sides can form a triangle.