Similar -Triangles are equal to the ratio of the squares of any two corresponding sides.

Area(ΔABC) AB^{2} BC^{2} AC^{2}----------- = ---------= -------- = ------- Area(ΔDEF) DE ^{2} EF^{2} DF^{2} |

1) The areas of two Similar-Triangles are in the ratio of the squares of the corresponding altitudes. 2) The areas of two Similar-Triangles are in the ratio of the squares of the corresponding medians. 3) The areas of two similar-triangles are in the ratio of the squares of the corresponding angle bisector segments. 4) If the areas of two similar-triangles are equal, then the triangles are congruent .i.e equal and similar triangles are congruent. |

1) If ΔABC ~ ΔDEF such that BC = 3 cm and EF = 4 cm and the

area of ΔABC = 54 cm

Since the ratio of the areas of two similar triangles are equal to the ratio of the squares of any two corresponding sides.

Area(ΔABC) BC^{2}------------ = ------ Area(ΔDEF) EF ^{2} |

⇒ Area (Δ DEF ) = ( 54 x 16 ) / 9

⇒ Area ( Δ DEF) = 96 cm

2) Two isosceles triangles have equal vertical angles and their areas are in the ratio 16 : 25. Find the ratio of their corresponding heights.

Area(ΔABC) 16 -------------- = ------- Area(ΔDEF) 25 |

Now, AB = AC , DE = DF

⇒ AB / AC = 1 and DE / DF = 1

⇒ AB / AC = DE / DF

⇒ AB / DE = AC / DF ( By alternendo)

Thus, in triangles ABC and DEF, we have,

AB / DE = AC / DF and ∠A = ∠ D

ΔABC ~ ΔDEF ( By SAS )

⇒ Area ( Δ ABC ) / Area ( ΔDEF ) = AL

⇒ 16 / 25 = AL

⇒ AL / DM = 4 /5

Hence, AL : DM = 4 : 5

• Similarity in Geometry

• Properties of similar triangles

• Basic Proportionality Theorem(Thales theorem)

• Converse of Basic Proportionality Theorem

• Interior Angle Bisector Theorem

• Exterior Angle Bisector Theorem

• Proofs on Basic Proportionality

• Criteria of Similarity of Triangles

• Geometric Mean of Similar Triangles

• Areas of Two Similar Triangles

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