Geometry:area-similartriangles

# Area-SimilarTriangles

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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

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