Here we will prove that if a perpendicular is drawn from the rightangled vertex of rightangled triangle to the hypotenuse and if the sides of the rightangled 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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