1. Prove pythagoras theorem using similarity.
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Yogita Ingle 5 years, 3 months ago
Pythagoras’ Theorem: In a right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides.
Given: Let XYZ be a triangle in which ∠YXZ = 90°.
YZ is the hypotenuse.
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To prove: XY2 + XZ2 = YZ2.
Construction: Draw XM ⊥YZ.
Therefore, ∠XMY = ∠XMZ = 90°.
Proof:
Statement
Reason
1. In ∆XYM and ∆XYZ,
(i) ∠XMY = ∠YXZ = 90°
(ii) ∠XYM = ∠XYZ
1.
(i) Given and by construction
(ii) Common angle
2. Therefore, ∆XYM ∼ ∆ZYX
2. BY AA criterion of similarity
3. Therefore, XYYZXYYZ = YMXYYMXY
3. Corresponding sides of similar triangle are proportional
4. Therefore, XY22 = YZ ∙ YM
4. By cross multiplication in statement 3.
5. In ∆XMZ and ∆XYZ,
(i) ∠XMY = ∠YXZ = 90°
(ii) ∠XZM = ∠XZY
5.
(i) Given and by construction
(ii) Common angle
6. Therefore, ∆XMZ ∼ ∆YXZ.
6. BY AA criterion of similarity
7. Therefore, XZ/YZ/ = MZ/XZ
7. Corresponding sides of similar triangle are proportional
8. Therefore, XZ2 = YZ ∙ MZ
8. By cross multiplication in statement 7.
9. Therefore, XY2 + XZ2 = YZ ∙ YM + YZ ∙ MZ
⟹ XY2 + XZ2 = YZ(YM+ MZ)
⟹ XY2 + XZ2 = YZ ∙ YZ
⟹ XY2 + XZ2= YZ2
9. By adding statements 4 and 8
0Thank You