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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