Triangles

Triangles

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  TrianglesTriangles By Eisa, Devaand Rajat
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  • Basic ProportionalityTheorem • Similarity Criteria • Area Theorem • Pythagoras Theorem What will you learn?
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  • Basic ProportionalityTheorem states that if a line is drawn parallel to one side of a triangle to intersect the other 2 points , the other 2 sides are divided in the same ratio. • It was discovered by Thales , so also known as Thales theorem. Basic Proportionality Theorem
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  ProvingtheThales’Theorem
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  Converse of theThales’ Theorem If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side
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  ProvingtheconverseofThales’ Theorem
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  Similarity Criteria Similarity Criterias SSSASA AA
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  • If intwo triangles, corresponding angles are equal, then their corresponding sides are in the same ratio (or proportion) and hence the two triangles are similar. • In Δ ABC and Δ DEF if ∠ A=∠ D, ∠ B= ∠E and ∠ C =∠ F then Δ ABC ~ Δ DEF. AAA Similarity
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  • If intwo triangles, sides of one triangle are proportional to (i.e., in the same ratio of ) the sides of the other triangle, then their corresponding angles are equal and hence the two triangles are similar. • In Δ ABC and Δ DEF if AB/DE =BC/EF =CA/FD then Δ ABC ~ Δ DEF. SSS Similarity
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  • If oneangle of a triangle is equal to one angle of the other triangle and the sides including these angles are proportional, then the two triangles are similar. • In Δ ABC and Δ DEF if AB/DE =BC/EF and ∠ B= ∠E then Δ ABC ~ Δ DEF. SAS Similarity
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  • The ratioof the areas of two similar triangles is equal to the square of the ratio of their corresponding sides • It proves that in the figure given below Area Theorem
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  ProofofAreaTheorem
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  • If aperpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse then triangles on both sides of the perpendicular are similar to the whole triangle and to each other • In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. • In a right triangle if a and b are the lengths of the legs and c is the length of hypotenuse, then a² + b² = c². • It states Hypotenuse² = Base² + Altitude². • A scientist named Pythagoras discovered the theorem, hence came to be known as Pythagoras Theorem. Pythagoras Theorem
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  Pythagoras Theorem
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  ProofofPythagorasTheorem
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  In a triangle,if square of one side is equal to the sum of the squares of the other two sides, then the angle opposite the first side is a right angle. Converse of Pythagoras Theorem
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  ProofofConverseofPythagorasTheorem
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  • Two figureshaving the same shape but not necessarily the same size are called similar figures. • All the congruent figures are similar but the converse is not true. • Two polygons of the same number of sides are similar, if (i) their corresponding angles are equal and (ii) their corresponding sides are in the same ratio (i.e., proportion). • If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then the other two sides are divided in the same ratio. • If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side. Summary
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  • If intwo triangles, corresponding angles are equal, then their corresponding sides are in the same ratio and hence the two triangles are similar (AAA similarity criterion). • If in two triangles, two angles of one triangle are respectively equal to the two angles of the other triangle, then the two triangles are similar (AA similarity criterion). • If in two triangles, corresponding sides are in the same ratio, then their corresponding angles are equal and hence the triangles are similar (SSS similarity criterion). • If one angle of a triangle is equal to one angle of another triangle and the sides including these angles are in the same ratio (proportional), then the triangles are similar (SAS similarity criterion). • The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. Summary
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  • If aperpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse, then the triangles on both sides of the perpendicular are similar to the whole triangle and also to each other. • In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides (Pythagoras Theorem). • If in a triangle, square of one side is equal to the sum of the squares of the other two sides, then the angle opposite the first side is a right angle. Summary
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  Thank you