1. Prove pythagoras theorem using similarity.

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Posted by Sumit Gupta 3 years, 5 months ago

Yogita Ingle 3 years, 5 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.

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