

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 90^{0} 
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) BC^{2}  =  Area(ΔABC) QR^{2} 
11) By property of multiplication 


1) Area(ΔABC) AB^{2}  =  Area(ΔDEF) DE^{2} 
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. 
AB^{2} AX^{2} 8)  =  DE^{2} DY^{2} 
8) Squaring both sides 
Area(ΔABC) AX^{2} 9)  =  Area(ΔDEF) DY^{2} 
9) From (1) and (8) 