Prove that if a perpendicular is drawn from the right angle vertex of a right-angled triangle to the hypotenuse, then the two adjacent triangles formed are similar to each other and each is also similar to the original triangle.

Q.Prove that if a perpendicular is drawn from the right angle vertex of a right-angled triangle to the hypotenuse, then the two adjacent triangles formed are similar to each other and each is also similar to the original triangle.

Given: ABC is a right-angled triangle with ∠A = 90°, and a perpendicular AD is drawn from the right angle vertex A to the hypotenuse BC. To Prove: Triangles ∆DBA and ∆DAC are similar to each other. Proof: In ∆DBA and ∆ABC: ∠BDA = ∠BAC = 90° And ∠ABD = ∠CBA Therefore, the remaining angle ∠BAD = ∠BCA So, ∆DBA and ∆ABC are equiangular Therefore, ∆DBA is similar to ∆ABC Similarly, in ∆DAC and ∆ABC: ∠ADC = ∠BAC = 90° ∠ACD = ∠BCA Therefore, the remaining angle ∠CAD = ∠CBA So, ∆DAC and ∆ABC are equiangular Therefore, ∆DAC is similar to ∆ABC Now, since both ∆DBA and ∆DAC are similar to ∆ABC, Therefore, ∆DBA is similar to ∆DAC (Proved)

Left Side	Right Side
(a) A parallelogram inscribed in a circle	(i) Proportional
(b) If two circles touch internally, the distance between their centers is —	(ii) Right angle
(c) The corresponding sides of two similar-angled triangles are	(iii) Rectangle
(d) An angle in a semicircle	(iv) Difference in radii