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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 |
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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) |