Question

If a perpendicular is drawn form the right angle of a right angled triangle to the hypotenuse,prove that the triangles on each side of the perpendicular are similar to the whole triangle and to each other​

Answer 1

Answer:

Given : ∆ ABC right angled at B

and perpendicular from intersecting AC at D ( BD perpendicular AC)

To prove : ∆ ADB ~ ∆ ABC

∆ BDC ~ ∆ ABC

∆ ADB ~ ∆ BDC

Proof : In ∆ ADB and ∆ ABC

angle A = angle A ( common)

angle ADB = angle ABC ( 90 ° each)

(by AA similarity criterion)

∆ ADB ~ ∆ ABC ..........from 1

Similarly

∆ BDC and ∆ ABC

angle C = angle C ( COMMON)

angle BDC = angle ABC ( EACH 90 °)

(BY AA similarity criterion)

∆ BDC ~ ∆ ABC ........from 2

from 1 and 2

∆ ADB ~ ∆ ABC and ∆ BDC ~ ∆ ABC

IF ONE TRIANGLE SIMILAR to another triangle, and second triangle is similar to third triangle

then first and third similar

therefore ∆ ADB ~ ∆ BDC