Area-SimilarTriangles

In this section, we will show you how to solve proofs on area-similartriangles.

Theorem 1: The ratio of the areas of two similar triangles are equal to the ratio of the squares of any two corresponding sides.

Given : Two triangles ABC and DEF such that ΔABC ~ ΔDEF.

Prove that : Area(ΔABC)/Area(ΔDEF) = AB 2 /DE 2 = BC 2 /EF 2 = AC 2 /DF 2

Construction : Draw AM ⊥ BC and PN ⊥ QR

Statements
Reasons
1) ΔABC ~ ΔPQR 1) Given
2) AB BC AC
------- = -------- = -------
PQ QR PR
2) If two Δ's are similar then their corresponding sides are in the ratio
3) ∠ABM = ∠PQN 3) As ΔABC ~ ΔPQR
4) ∠AMB = ∠PNQ 4) Each of measure 900
5) ΔAMB ~ ΔPNQ 5) By AA similarity
6) AM PN
------- = --------
AB PQ
6) If two Δ's are similar then their corresponding sides are in the ratio
7) AM PN
------- = --------
BC QR
7) From (2) and (6)
8) Area(ΔABC) (Base x height)/2
-------------- = -------------------
Area(ΔABC) (Base x height)/2
8) By definition of Area
9) Area(ΔABC) (BC x AN)
----------- = ---------------
Area(ΔABC) (QR x PN)
9) By substitution
10) Area(ΔABC) (BC x BC)
------------- = --------------
Area(ΔABC) (QR x QR)
10) By substitution
11) Area(ΔABC) BC2
---------- = ----------
Area(ΔABC) QR2
11) By property of multiplication


Theorem 2 on area-similartriangles : The area of two similar triangles are in the ratio of the squares of the corresponding angle bisector segments.

Proof :
Given ΔABC ~ ΔDEF and AX and DY are the bisectors of ∠A and ∠D respectively.
To prove that : [area(ΔABC)/area(ΔDEF)] = (AX) 2 /(DY) 2

Statements
Reasons
1) Area(ΔABC) AB2
--------------- = ----------
Area(ΔDEF) DE2
1) The ratio of the areas of two similar triangles are equal to the ratio of the squares of any two corresponding sides.
2) ∠A = ∠D 2) As the two triangles are similar
3) 1/2∠A = 1/2∠D 3) Multiply both sides by 1/2
4) ∠BAX = ∠EDY 4) By definition of angle bisector
5) ∠B = ∠E 5) ΔABC ~ ΔDEF
6) ΔABX ~ ΔDEY 6) By AA criteria or rule
AB AX
7) ------- = --------
DE DY
7)If two triangles are similar then their corresponding sides are in ratio.
AB2 AX2
8) ------- = --------
DE2 DY2
8) Squaring both sides
Area(ΔABC) AX2
9) --------------- = ----------
Area(ΔDEF) DY2
9) From (1) and (8)

Geometry proofs
GeometryProof-1
GeometryProof 2
area-similartriangles
Pythagorean theorem

Home