Class 7 Mathematics Notes Chapter 6 (The Triangle and its Properties) – Mathematics Book

Alright class, let's get straight into Chapter 6: The Triangle and its Properties. This is a foundational chapter, and understanding these concepts thoroughly is crucial not just for your class exams but also for various government competitive exams where basic geometry questions are common. Pay close attention!

Chapter 6: The Triangle and its Properties - Detailed Notes

1. Introduction to Triangles

• Definition: A triangle is a simple closed curve made of three line segments.
• Elements: A triangle has:
  • Three Sides: Line segments forming the triangle (e.g., AB, BC, CA in ΔABC).
  • Three Angles: Formed at the vertices where sides meet (e.g., ∠A, ∠B, ∠C or ∠BAC, ∠ABC, ∠BCA).
  • Three Vertices: Points where the sides meet (e.g., A, B, C).
• Notation: A triangle with vertices A, B, and C is denoted as ΔABC.

2. Classification of Triangles

Triangles can be classified based on their sides and angles:

• (a) Based on Sides:

  • Scalene Triangle: All three sides are of different lengths. All three angles are also usually different.
  • Isosceles Triangle: Any two sides are of equal length. The angles opposite the equal sides are also equal.
  • Equilateral Triangle: All three sides are of equal length. All three angles are equal, and each measures 60°.

• (b) Based on Angles:

  • Acute-angled Triangle: All three angles are acute (less than 90°).
  • Obtuse-angled Triangle: One angle is obtuse (greater than 90°). A triangle can have only one obtuse angle.
  • Right-angled Triangle: One angle is a right angle (exactly 90°). The side opposite the right angle is called the hypotenuse, and it is the longest side. The other two sides are called legs.

3. Medians of a Triangle

• Definition: A median connects a vertex of a triangle to the mid-point of the opposite side.
• Properties:
  • Every triangle has three medians, one from each vertex.
  • The three medians always intersect at a single point inside the triangle called the Centroid.
  • A median divides the triangle into two triangles of equal area.

4. Altitudes of a Triangle

• Definition: An altitude is the perpendicular line segment from a vertex to its opposite side (or the extension of the opposite side). It represents the 'height' of the triangle from that vertex.
• Properties:
  • Every triangle has three altitudes, one from each vertex.
  • The three altitudes intersect at a single point called the Orthocenter.
  • The orthocenter can lie inside (for acute-angled triangles), outside (for obtuse-angled triangles), or on the vertex of the right angle (for right-angled triangles).

5. Exterior Angle of a Triangle and its Property

• Definition: An exterior angle is formed when one side of a triangle is extended. It is adjacent to an interior angle and forms a linear pair (sum = 180°) with it.
• Property: The measure of an exterior angle of a triangle is equal to the sum of the measures of its two interior opposite angles.
  • Example: In ΔABC, if side BC is extended to D, then Exterior ∠ACD = Interior ∠A + Interior ∠B.
• Corollary: An exterior angle is always greater than either of its interior opposite angles.

6. Angle Sum Property of a Triangle

• Property: The sum of the measures of the three interior angles of any triangle is always 180°.
  • In ΔABC, ∠A + ∠B + ∠C = 180°.
• Application: If two angles of a triangle are known, the third angle can be calculated using this property.

7. Special Properties of Isosceles and Equilateral Triangles

• Isosceles Triangle Property:
  • If two sides of a triangle are equal, then the angles opposite to these sides are also equal.
  • Conversely, if two angles of a triangle are equal, then the sides opposite to these angles are also equal.
• Equilateral Triangle Property:
  • All sides are equal.
  • All angles are equal to 60°.

8. Triangle Inequality Property

• Property: The sum of the lengths of any two sides of a triangle is always greater than the length of the third side.
  • In ΔABC:
    • AB + BC > AC
    • BC + AC > AB
    • AC + AB > BC
• Significance: This property helps determine if three given lengths can form a triangle. If the sum of the two smaller lengths is greater than the largest length, a triangle can be formed.
• Corollary: The difference between the lengths of any two sides of a triangle is always less than the length of the third side.

9. Right-angled Triangles and Pythagoras Theorem

• Definition: A triangle with one angle equal to 90°.
• Pythagoras Theorem: In a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides (legs).
  • If ∠B = 90° in ΔABC, then AC is the hypotenuse.
  • Formula: (Hypotenuse)² = (Leg1)² + (Leg2)² or AC² = AB² + BC²
• Converse of Pythagoras Theorem: If the square of one side of a triangle is equal to the sum of the squares of the other two sides, then the angle opposite the first side is a right angle.
• Application: Used to find the length of an unknown side in a right-angled triangle when the other two sides are known. Also used to check if a triangle is right-angled.

Multiple Choice Questions (MCQs)

Here are 10 MCQs based on the chapter for your practice:

1. If the angles of a triangle are 45°, 65°, and 70°, the triangle is:
   (a) Obtuse-angled
   (b) Right-angled
   (c) Acute-angled
   (d) Isosceles

2. In ΔPQR, side QR is extended to S. If ∠P = 50° and ∠Q = 70°, what is the measure of exterior ∠PRS?
   (a) 70°
   (b) 120°
   (c) 110°
   (d) 60°

3. Which of the following sets of side lengths can form a triangle?
   (a) 3 cm, 4 cm, 7 cm
   (b) 5 cm, 6 cm, 12 cm
   (c) 6 cm, 8 cm, 10 cm
   (d) 2 cm, 3 cm, 6 cm

4. In a right-angled triangle ABC, if ∠B = 90°, AB = 5 cm, and BC = 12 cm, what is the length of the hypotenuse AC?
   (a) 17 cm
   (b) 7 cm
   (c) 13 cm
   (d) 10 cm

5. A line segment connecting a vertex of a triangle to the mid-point of the opposite side is called:
   (a) An altitude
   (b) A median
   (c) An angle bisector
   (d) A perpendicular bisector

6. In an isosceles triangle, one of the equal angles measures 55°. What is the measure of the third angle?
   (a) 55°
   (b) 70°
   (c) 80°
   (d) 125°

7. How many altitudes can a triangle have?
   (a) 1
   (b) 2
   (c) 3
   (d) Infinite

8. The sum of the three angles of any triangle is always equal to:
   (a) 90°
   (b) 180°
   (c) 270°
   (d) 360°

9. In ΔXYZ, if XY = YZ and ∠Y = 80°, what is the measure of ∠X?
   (a) 80°
   (b) 100°
   (c) 40°
   (d) 50°

10. The point of concurrence (intersection) of the medians of a triangle is called the:
    (a) Orthocenter
    (b) Incenter
    (c) Circumcenter
    (d) Centroid

Answers to MCQs:

1. (c) - All angles are less than 90°.
2. (b) - Exterior angle = Sum of interior opposite angles = 50° + 70° = 120°.
3. (c) - Triangle Inequality: 6+8 > 10 (14>10), 6+10 > 8 (16>8), 8+10 > 6 (18>6). Also, it's a Pythagorean triplet (6²+8²=36+64=100=10²), forming a right triangle.
4. (c) - Pythagoras Theorem: AC² = AB² + BC² = 5² + 12² = 25 + 144 = 169. AC = √169 = 13 cm.
5. (b) - Definition of a median.
6. (b) - The equal angles are 55° each. Third angle = 180° - (55° + 55°) = 180° - 110° = 70°.
7. (c) - One altitude from each vertex.
8. (b) - Angle Sum Property.
9. (d) - Since XY = YZ, ∠X = ∠Z (angles opposite equal sides). Let ∠X = ∠Z = x. By Angle Sum Property: x + 80° + x = 180° => 2x = 100° => x = 50°. So, ∠X = 50°.
10. (d) - Definition of Centroid.

Revise these notes thoroughly. Remember the definitions, properties, and theorems, especially the Angle Sum Property, Exterior Angle Property, Triangle Inequality, and Pythagoras Theorem, as they are frequently tested. Good luck with your preparation!