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) = AB²/DE² = BC²/EF² = AC²/DF²

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)

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

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