# Greater segment of the Hypotenuse is Equal to the Smaller Side of the Triangle

Here we will prove that if a perpendicular is drawn from the right-angled vertex of right-angled triangle to the hypotenuse and if the sides of the right-angled triangle are in continued proportion, the greater segment of the hypotenuse is equal to the smaller side of the triangle.

Solution:

In ∆ XYZ, ∠XYZ = 90°. YP ⊥ XZ.

XY < YZ and YZ < XZ.

Also $$\frac{XY}{YZ}$$ = $$\frac{YZ}{XZ}$$

To prove: XY = PZ.

Proof:

 Statement Reason 1. ∆ XYZ and ∆ YPZ,(i) ∠XZY = ∠PZY(ii) ∠XYZ = ∠YPZ = 90°. 1.(i) Common angle.(ii) Given. 2. ∆ XYZ ∼ ∆ YPZ. 2.  By AA criterion of similarity. 3. Therefore, $$\frac{YZ}{XZ}$$ = $$\frac{PZ}{YZ}$$. 3.  Corresponding sides of similar triangles are proportional. 4. But, $$\frac{XY}{YZ}$$ = $$\frac{YZ}{XZ}$$. 4. Given. 5. Therefore, $$\frac{XY}{YZ}$$ = $$\frac{PZ}{YZ}$$. 5. From statements 3 and 4. 6. Therefore, XY = PZ. (Proved) 6. From statement 5.

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