Triangle
°Intro:
Hello Everybody! Today, we are going to discuss the triangles and their theorems. Triangles are one of the most fundamental shapes in geometry, and they play a crucial role in various aspects of mathematics, architecture, engineering, and daily life. In Chapter 7 of the Class 9 NCERT book, you will dive into the properties of triangles and different theorems associated with them. Understanding these properties is essential for building a strong foundation in geometry.
1. What is a Triangle?
A triangle is a closed figure with three sides, three angles, and three vertices.
A triangle is a polygon with the smallest number of sides.
The sum of the three interior angles of any triangle is always 180°.
A triangle is denoted by (∆ ABC ), where ( A ), ( B ), and ( C ) are the vertices of the triangle.
2. Classification of Triangles:
Triangles can be classified based on their sides and angles:
– Based on Sides:
1. Scalene Triangle: A triangle with all three sides of different lengths.
2. Isosceles Triangle: A triangle with two sides of equal length.
3. Equilateral Triangle: A triangle with all three sides of equal length. In an equilateral triangle, all angles are also equal, and each angle measures (60°).
– Based on Angles:
1. Acute-angled Triangle: A triangle where all three angles are less than (90°).
2. Right-angled Triangle: A triangle where one angle is exactly (90°). The side opposite the right angle is called the hypotenuse.
3. Obtuse-angled Triangle:A triangle where one of the angles is greater than (90°).
3. Important Properties of Triangles
– Property 1: Angle Sum Property of a Triangle:
The sum of the interior angles of a triangle is always (180° ). This can be stated as:
∠A + ∠B + ∠C = 180°
This property is true for any type of triangle.
– Property 2: Exterior Angle Property of a Triangle:
The exterior angle of a triangle is equal to the sum of the two interior opposite angles. For a triangle ( ∆ ABC ), if an exterior angle ( ∆ ACD ) is formed at vertex (C), then:
∠ACD = ∠A + ∠B
This property helps in solving many problems related to triangle angles.
– Property 3: Triangle Inequality Theorem:
The sum of the lengths of any two sides of a triangle is always greater than the length of the third side.
For a triangle with sides ( a ), ( b ), and ( c ):
a + b > c, a + c > b, b + c > a
This theorem is useful in determining whether a set of three given sides can form a triangle or not.
– Property 4: Pythagoras Theorem (for Right-Angled Triangles):
In a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides. For a right-angled triangle (∆ ABC ) where ( ∠B = 90° ):
BC ² = AB² + AC² or BC = √(AB² + AC²)
(Hypotenuse)² = (Base)² + ( Perpendicular)²
This theorem is one of the most important results in geometry, and it has numerous applications.
4. Congruence of Triangles:
Two triangles are said to be congruent if all corresponding sides and angles are equal. Congruence is denoted by the symbol ( ≅ ). If two triangles are congruent, they are essentially identical in shape and size.
– Criteria for Congruence of Triangles:
There are several rules or criteria to check whether two triangles are congruent:
1. SSS (Side-Side-Side) Criterion: If the three sides of one triangle are equal to the three sides of another triangle, then the two triangles are congruent. Look at the triangles below, the triangles are said to be congruent because AC = ZY, CB = ZX, and AB = XY. Hence, ∆ABC ≅ ∆XYZ.
2. SAS (Side-Angle-Side) Criterion: If two sides and the angle between them in one triangle are equal to two sides and the included angle of another triangle, then the triangles are congruent.
3. ASA (Angle-Side-Angle) Criterion: If two angles and the side between them in one triangle are equal to the corresponding two angles and the included side of another triangle, then the triangles are congruent.
4. AAS (Angle-Angle-Side) Criterion: If two angles and a non-included side of one triangle are equal to the corresponding two angles and a non-included side of another triangle, then the triangles are congruent.
5. RHS (Right Angle-Hypotenuse-Side) Criterion: In right-angled triangles, if the hypotenuse and one side of one triangle are equal to the hypotenuse and one side of another right-angled triangle, then the triangles are congruent.
5. Some Important Theorems Related to Triangles:
Theorem 1: The Angle Sum Property of a Triangle:
As discussed earlier, the sum of the angles of a triangle is (180°). This can be used to find the unknown angle in a triangle if two angles are known.
Theorem 2: Exterior Angle Theorem:
The exterior angle of a triangle is equal to the sum of the two interior opposite angles.
Theorem 3: Converse of the Pythagoras Theorem:
If in a triangle, the square of one side is equal to the sum of the squares of the other two sides, then the triangle is a right-angled triangle.
Theorem 4: Inequality in a Triangle:
For any triangle, the sum of any two sides of a triangle is greater than the third side.
6. Medians and Altitudes of a Triangle:
– Median of a Triangle:
A median of a triangle is a line segment that joins a vertex to the midpoint of the opposite side. Every triangle has three medians, and they all intersect at a point called the centroid of the triangle. The centroid divides each median into two parts such that the ratio of the longer part to the shorter part is 2:1.
– Altitude of a Triangle:
An altitude of a triangle is a perpendicular line segment from a vertex to the opposite side (or the line containing the opposite side). Every triangle has three altitudes, and they intersect at a point called the orthocenter.
7. Properties of Isosceles and Equilateral Triangles
– Isosceles Triangle Properties:
Two sides of equal length.
Two angles opposite the equal sides are also equal.
The altitude from the vertex angle to the base is also a median and an angle bisector.
– Equilateral Triangle Properties:
All three sides are equal.
All three angles are equal, and each angle is (60°).
The altitudes, medians, angle bisectors, and perpendicular bisectors all coincide at a single point called the centroid.
°8. Application of Triangles in Real Life:
– Construction and Architecture: Triangles are used in the design of buildings and bridges to provide stability and distribute weight efficiently.
– Engineering: In engineering, especially in mechanical structures, the stability of objects often depends on triangular components.
– Astronomy and Geography: Triangulation is a method used in astronomy and geography to measure distances using triangles.
– Surveying: Surveyors use triangulation to map out areas and determine distances between points.
°Hi Shortly:
The study of triangles is foundational in geometry and has vast applications in real life. By mastering the properties, theorems, and methods for solving problems related to triangles, students build essential problem-solving skills that are useful in advanced geometry and other branches of mathematics. Through this chapter, we explore not only the beauty of geometric reasoning but also the practical implications of triangles in our everyday world.
– A triangle is a polygon with three sides, three angles, and three vertices.
– The sum of the interior angles of a triangle is always (180°).
– The exterior angle of a triangle is equal to the sum of the opposite interior angles.
– There are different criteria for the congruence of triangles, such as SSS, SAS, ASA, AAS, and RHS.
– The Pythagoras Theorem holds for right-angled triangles, and it is widely used in real-world problems.
– Triangles are classified based on their sides and angles, such as scalene, isosceles, and equilateral triangles.
– Medians, altitudes, and the centroid play crucial roles in understanding triangle properties.