The Triangle and Its Properties: Class 7 Maths NCERT Guide

Welcome to the fascinating world of triangles! A triangle is one of the most basic and important shapes in geometry. It's a simple closed figure made of three line segments. But don't let its simplicity fool you! Triangles have some amazing properties that are used everywhere, from building strong bridges and tall buildings to creating art and video games. In this chapter, 'The Triangle and Its Properties,' we will explore these special rules. You will learn about the different parts of a triangle, like medians and altitudes. We'll uncover the secrets of its angles with the Exterior Angle Property and the Angle Sum Property. We will also learn about the famous Pythagoras theorem, a powerful tool for right-angled triangles. By the end of this guide, you will be able to confidently solve problems involving triangles and understand why they are so fundamental to mathematics.

Elements and Types of Triangles

A triangle is a polygon with three sides, three angles, and three vertices. Let's name a triangle as ΔABC. The three vertices are the points A, B, and C. The three sides are the line segments AB, BC, and CA. The three angles are ∠BAC, ∠ABC, and ∠BCA (or simply ∠A, ∠B, and ∠C).

We can classify triangles in two main ways:

1. Based on Sides:

  • Equilateral Triangle: All three sides are equal in length. (AB = BC = CA). All angles are also equal (60° each).
  • Isosceles Triangle: Any two sides are equal in length. (e.g., AB = AC). The angles opposite the equal sides are also equal.
  • Scalene Triangle: All three sides are of different lengths. All three angles are also different.

2. Based on Angles:

  • Acute-angled Triangle: All three angles are acute (less than 90°).
  • Right-angled Triangle: One angle is a right angle (exactly 90°).
  • Obtuse-angled Triangle: One angle is an obtuse angle (greater than 90°).

Understanding Medians and Altitudes

Two very important lines you can draw inside a triangle are the median and the altitude. It's easy to get them confused, so let's look closely.

Median:
A median is a line segment that connects a vertex of a triangle to the midpoint of the opposite side. Think of 'median' like 'middle'. Since a triangle has three vertices, it has three medians. For example, in ΔABC, if you find the exact middle point of side BC and call it D, the line segment AD is a median. A key property is that all three medians of a triangle always meet at a single point inside the triangle, called the centroid.

Altitude:
An altitude is a line segment from a vertex that is perpendicular (makes a 90° angle) to the opposite side. Think of 'altitude' like 'height'. A triangle also has three altitudes. In ΔPQR, the line segment from vertex P that hits the side QR at a right angle is an altitude. The position of an altitude can change: in an acute triangle, it is inside; in a right-angled triangle, two of its sides are the altitudes; and in an obtuse triangle, the altitude from the obtuse angle vertex may lie outside the triangle!

Key Angle Properties with Examples

  • 1. Exterior Angle Property: The measure of an exterior angle of a triangle is equal to the sum of its two interior opposite angles.

    Example: In ΔXYZ, the side YZ is extended to a point P. If ∠X = 60° and ∠Y = 50°, find the exterior angle ∠XZP.
    Solution:

    • The interior opposite angles are ∠X and ∠Y.
    • According to the property, Exterior Angle = Sum of interior opposite angles.
    • ∠XZP = ∠X + ∠Y
    • ∠XZP = 60° + 50° = 110°.

  • 2. Angle Sum Property: The sum of the measures of the three interior angles of any triangle is always 180°.

    Example: In ΔABC, if ∠A = 75° and ∠B = 45°, find ∠C.
    Solution:

    • We know that ∠A + ∠B + ∠C = 180°.
    • Substitute the known values: 75° + 45° + ∠C = 180°.
    • First, add the known angles: 120° + ∠C = 180°.
    • Now, find ∠C by subtracting from 180°: ∠C = 180° - 120° = 60°.

How to Use the Pythagoras Theorem

  1. Step 1: Check for a Right Angle — The Pythagoras theorem works ONLY for right-angled triangles. Identify the angle that is exactly 90°.
  2. Step 2: Identify the Hypotenuse — The side opposite the 90° angle is called the hypotenuse. It is always the longest side of the triangle. The other two sides are called the legs.
  3. Step 3: Apply the Formula — The theorem states: (leg 1)² + (leg 2)² = (hypotenuse)². If the sides are a, b, and the hypotenuse is c, the formula is a² + b² = c².
  4. Step 4: Solve for the Unknown Side — Let's find the hypotenuse of a right-angled triangle whose legs are 6 cm and 8 cm. - Using the formula: 6² + 8² = c² - Calculate the squares: 36 + 64 = c² - Add them up: 100 = c² - Find the square root: c = √100 = 10 cm. The hypotenuse is 10 cm.

Important Tips for Your Exam

1. Don't Confuse Median and Altitude: Remember, a Median connects to the Midpoint, while an Altitude forms a 90° angle (height). They are not always the same line!

2. Pythagoras is Only for Right-Angled Triangles: Never try to use a² + b² = c² on an acute or obtuse triangle. It won't work.

3. Triangle Inequality Property: Don't forget this simple but powerful rule! The sum of the lengths of any two sides of a triangle must be greater than the length of the third side. For example, you cannot form a triangle with sides 3 cm, 4 cm, and 8 cm because 3 + 4 is not greater than 8.

Practice Questions with Solutions

  • Q: The two interior opposite angles of an exterior angle of a triangle are 65° and 45°. Find the measure of the exterior angle.
    A: Step 1: Recall the Exterior Angle Property of a triangle. It states that the exterior angle is equal to the sum of the two interior opposite angles.
    Step 2: Identify the given interior opposite angles: 65° and 45°.
    Step 3: Add these two angles to find the exterior angle: 65° + 45° = 110°.
    Final answer: The measure of the exterior angle is 110°.

  • Q: In a triangle ΔPQR, ∠P = 30° and ∠Q = 40°. Find the measure of ∠R.
    A: Step 1: Recall the Angle Sum Property of a triangle, which states that the sum of all three interior angles is 180°.
    Step 2: Write down the equation: ∠P + ∠Q + ∠R = 180°.
    Step 3: Substitute the given values: 30° + 40° + ∠R = 180°.
    Step 4: Simplify the equation: 70° + ∠R = 180°.
    Step 5: Solve for ∠R: ∠R = 180° - 70° = 110°.
    Final answer: The measure of ∠R is 110°.

  • Q: A 15 m long ladder reached a window 12 m high from the ground on placing it against a wall. Find the distance of the foot of the ladder from the wall.
    A: Step 1: Visualize the problem. The ladder, the wall, and the ground form a right-angled triangle. The ladder is the hypotenuse (15 m), and the height of the window on the wall is one leg (12 m). We need to find the other leg (distance from the wall).
    Step 2: Apply the Pythagoras theorem: a² + b² = c². Here, c = 15, a = 12, and we need to find b.
    Step 3: Substitute the values: 12² + b² = 15².
    Step 4: Calculate the squares: 144 + b² = 225.
    Step 5: Solve for b²: b² = 225 - 144 = 81.
    Step 6: Find the square root: b = √81 = 9.
    Final answer: The distance of the foot of the ladder from the wall is 9 m.

  • Q: Can you have a triangle with sides of length 5 cm, 7 cm, and 13 cm?
    A: Step 1: Recall the Triangle Inequality Property. The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
    Step 2: Check all three combinations:

    • Is 5 + 7 > 13? 12 > 13. This is False.
    • Is 5 + 13 > 7? 18 > 7. This is True.
    • Is 7 + 13 > 5? 20 > 5. This is True.

    Step 3: Since one of the conditions (5 + 7 > 13) is false, a triangle cannot be formed with these side lengths.
    Final answer: No, a triangle cannot be formed with these side lengths because the sum of the two smaller sides (12 cm) is not greater than the largest side (13 cm).

Frequently Asked Questions

What is the main difference between a median and an altitude?

A median connects a vertex to the midpoint of the opposite side, so it bisects the side. An altitude connects a vertex to the opposite side at a 90° angle, representing the height of the triangle.

Can a triangle have two right angles?

No, a triangle cannot have two right angles. The sum of two right angles would be 90° + 90° = 180°. Since the sum of all three angles in a triangle must be 180°, this would leave 0° for the third angle, which is impossible.

What is the Triangle Inequality Property?

The Triangle Inequality Property states that the sum of the lengths of any two sides of a triangle must always be greater than the length of the third side. This rule determines if a set of three lengths can form a triangle.

How do I know which side is the hypotenuse in a right-angled triangle?

The hypotenuse is always the side directly opposite the 90° angle. It is also the longest side of the right-angled triangle.