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 \(\frac{XY}{YZ}\) = \(\frac{YZ}{XZ}\)

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

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.





9th Grade Math

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