Similarity Related To Right Triangle


 
 
Concept Explanation
 

Similarity Related To Right Triangle

Similarity Related To Right Triangle :

Theorem: If a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse then triangles on both sides of the perpendicular are similar to the whole triangle and to each other.

Given: A right triangle ABC right angled at B. BD is perpendicular to AC

To Prove: large (1);Delta ADB simDelta ABC

                  large (2) ;Delta BDC simDelta ABC

                 large (3);Delta ADB simDelta BDC

Proof:  large In;Delta ADB ;and;Delta ABC

                large angle A =angle A                 [Common]

               large angle D =angle B                [ Each large 90^0 ]

large therefore ;Delta ADB simDelta ABC              [By AA Similarity of Triangle]

Hence (1) statement has been proved.

  large In;Delta BDC ;and;Delta ABC

                large angle C =angle C                 [Common]

               large angle D =angle B                [ Each large 90^0 ]

large therefore ;Delta BDC simDelta ABC              [By AA Similarity of Triangle]

Hence (2) statement has been proved.

From statement (1) and (2) we see that both large ;Delta BDC ;and;Delta ADB;are;sim;Delta ABC

Therefore as both are similar to same triangle, they are similar to each other.

Hence large Delta ADB simDelta BDC

Hence (3) statement has been proved.

Illustration: In a triangle ABC right angled at C. If a perpendicular is drawn from C to AB meating AB at D. Prove that

large frac{BC^2}{AC^2}= frac{BD}{AD}

Proof: large Delta ACD simDelta ABC

So, large frac{AD}{AC}=frac{AC}{AB}

large Rightarrow AC^2= AB ;X;AD                     ............(1)

 Similarly in large Delta BCD simDelta BAC

So, large frac{BC}{BA}=frac{BD}{BC}

large Rightarrow BC^2= BD ;X;BA                     ............(2)

From (1) and (2) we get

large frac{BC^2}{AC^2} = frac{BD ;X;BA}{AB;X;AD}= frac{BD}{AD}

Hence Proved

 

Sample Questions
(More Questions for each concept available in Login)
Question : 1

In the following figure, ABD is a triangle in which angle DAB=90^{0} and AC perp BD. Then which of the following statement is/are true.

Right Option : B
View Explanation
Explanation
Question : 2

If a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse then triangles on both sides of the perpendicular are similar to

 ________________________.

Right Option : C
View Explanation
Explanation
Question : 3

In the following figure, ABD is a triangle in which angle DAB=90^{0} and AC perp BD. Then which of the following is true.

Right Option : A
View Explanation
Explanation
 
Video Link - Have a look !!!
 
Language - English
 
 
 
Related Videos
Language - English
Language - English



Students / Parents Reviews [10]