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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