Areas of Two Similar Triangles

Areas of Two similar Triangles : The ratio of the areas of two
Similar -Triangles are equal to the ratio of the squares of any two corresponding sides.
Area(ΔABC) AB2 BC2 AC2
----------- = ---------= -------- = -------
Area(ΔDEF) DE2         EF2         DF2

Some important theorems :
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.

Some solved examples
1) If ΔABC ~ ΔDEF such that BC = 3 cm and EF = 4 cm and the
area of ΔABC = 54 cm 2 . Find the area of ΔDEF.
 Solution :
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) BC2
------------ = ------
Area(ΔDEF)        EF2
⇒ 54 / [ Area(ΔDEF)] = 3 2 / 4 2
⇒ Area (Δ DEF ) = ( 54 x 16 ) / 9
⇒ Area ( Δ DEF) = 96 cm 2
2) Two isosceles triangles have equal vertical angles and their areas are in the ratio 16 : 25. Find the ratio of their corresponding heights.
 Solution : Let ΔABC and ΔDEF be the given triangles such that AB = AC and DE = DF, ∠ A = ∠ D.
Area(ΔABC) 16
-------------- = -------
Area(ΔDEF) 25
Construction : AL ⊥ BC and DM ⊥ EF
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 2 / DM 2
⇒ 16 / 25 = AL 2 / DM 2
⇒ AL / DM = 4 /5
Hence, AL : DM = 4 : 5
Similarity in Triangles

• 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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