Properties of a triangle

Triangle and its important properties

1. Of Triangle Properties

2. REMYA S 13003014 MATHEMATICS MTTC PATHANAPURAM

3. The Triangle and its Properties Triangle is a simple closed curve made of three line segments. Triangle has three vertices, three sides and three angles. In Δ ABC Sides : AB , BC and CA Angles : ∠ BAC , ∠ ABC and ∠ BCA Vertices : A, B and C The side opposite to the vertex A is BC.

4. Based on the sides Scalene Triangles No  equal sides No  equal angles Isosceles Triangles Two  equal sides  Two  equal angles Equilateral Triangles Three  equal sides Three  equal angles, always 60° Classification of triangles Scalene Isosceles Equilateral

5. Classification of triangles Based on Angles Acute-angled Triangle All angles are less than 90° Obtuse-angled Triangle Has an angle more than 90° Right-angled triangles Has a right angle (90°) Acute Triangle Right Triangle Obtuse Triangle

6. MEDIANS OF A TRIANGLE A median of a triangle is a line segment joining a vertex to the midpoint of the opposite side A triangle has three medians. The three medians always meet at a single point. Each median divides the triangle into two smaller triangles  which have the same area The centroid (point where they meet) is the center of gravity of the triangle .

7. ALTITUDES OF A TRIANGLE Altitude – line segment from a vertex that intersects the opposite side at a right angle. Any triangle has three altitudes.

8. Definition of an Altitude of a Triangle A segment is an altitude of a triangle if and only if it has one endpoint at a vertex of a triangle and the other on the line that contains the side opposite that vertex so that the segment is perpendicular to this line. ACUTE OBTUSE B A C ALTITUDES OF A TRIANGLE

9. RIGHT A B C If ABC is a right triangle, identify its altitudes. BG, AB and BC are its altitudes. G Can a side of a triangle be its altitude? YES! ALTITUDES OF A TRIANGLE

10. Proof: Ð C + Ð D + Ð E = 180 …….. Straight line Ð A = Ð D and Ð B = Ð E…. Alternate angles Þ Ð C + Ð B + Ð A = 180 Ð A + Ð B + Ð C = 180 D E Given: Triangle A B C Construction: Draw line ‘l’ through Ð C parallel to the base AB The measure of the three angles of a triangle sum to 180 . To Prove : Ð A + Ð B + Ð C = 180 l ANGLE SUM PROPERTY OF A TRIANGLE

11. An exterior angle of a triangle equals the sum of the two interior opposite angles in measure. To Prove: Ð ACD = Ð ABC + Ð BAC Proof: ÐA CB + Ð ACD = 180 0 …………………. Straight line Ð ABC + Ð ACB + Ð BAC = 180 ………………… sum of the triangle Þ Ð ACB + Ð ACD = Ð ABC + Ð ACB + Ð BAC Þ Ð ACD = Ð ABC + Ð BAC A B C D Given: In Δ ABC extend BC to D EXTERIOR ANGLE OF A TRIANGLE AND ITS PROPERTY

12. PYTHAGORAS THEOREM In a right angled triangle the square of the hypotenuse is equal to the sum of the squares of the other two sides. In ABC : AC is the hypotenuse AB and BC are the 2 sides Then according to Pythagoras theorem , A B C AC² = AB² + BC²