1.Diagonals AC and BD of a trapezium ABCD with AB parallel to DC intersect at point O. Using similarity criterion theorem for two triangles show that diagonals of AB parallel to DC intersect at point O.
 
2.Prove that "the areas of two similar triangles are equal they are not concurrent".
 
3.ABC is an isoceles triangle right angled at C. Prove that AB2 = 2AC.
 
4. In a rigt angled triangle the square of the hypotenuse is equal to the sum of the squares of the other two sides .
 
5.In triangle ABC if EF is parallel to AB and ar(triangle CEF) is congruent to ar(EFBA) then find the ratio of CA and EA. 

Asked by damarlaprathyusha40 | 2nd Oct, 2015, 10:37: AM

Expert Answer:

3. space G i v e n space t h a t space triangle A B C space i s space a n space i s o s c e l e s space r i g h t space t r i a n g l e space r i g h t space a n g l e d space a t space C. S i n c e space i t space i s space a n space i s o s c e l e s space t r i a n g l e space w i t h space C space a s space t h e space v e r t e x comma space w e space h a v e comma space A C equals B C. A l s o comma space triangle A B C space i s space a space r i g h t space t r i a n g l e space r i g h t space a n g l e d space a t space C. T h u s space b y space a p p l y i n g space P y t h a g o r a s space T h e o r e m comma space w e space h a v e comma A B squared equals A C squared plus B C squared S u b s t i t u t i n g space A C equals B C space i n space t h e space a b o v e space e q u a t i o n space w e space h a v e comma A B squared equals A C squared plus A C squared rightwards double arrow A B squared equals 2 A C squared 5. C o n s i d e r space t h e space f o l l o w i n g space f i g u r e.
G i v e n space t h a t space E F space i s space p a r a l l e l space t o space A B. rightwards double arrow fraction numerator C E over denominator C A end fraction equals fraction numerator C F over denominator C B end fraction rightwards double arrow fraction numerator C E over denominator C F end fraction equals fraction numerator C A over denominator C B end fraction rightwards double arrow triangle C E F tilde triangle A B C A l s o space g i v e n space t h a t space A r e a open parentheses triangle C E F close parentheses equals A r e a open parentheses triangle E F B A close parentheses rightwards double arrow space A r e a open parentheses triangle C E F close parentheses plus A r e a open parentheses triangle C E F close parentheses equals A r e a open parentheses triangle E F B A close parentheses plus A r e a open parentheses triangle C E F close parentheses rightwards double arrow 2 space A r e a open parentheses triangle C E F close parentheses equals A r e a open parentheses triangle C A B close parentheses fraction numerator space A r e a open parentheses triangle C E F close parentheses over denominator A r e a open parentheses triangle C A B close parentheses end fraction equals 1 half... left parenthesis 1 right parenthesis  S i n c e space triangle C E F space a n d space triangle A B C space a r e space s i m i l a r space t r i a n g l e s comma space t h e space r a t i o space o f space t h e i r space a r e a s space i s space t h e space s q u a r e space o f space t h e space r a t i o space o f space t h e i r space s i d e s. rightwards double arrow fraction numerator space A r e a open parentheses triangle C E F close parentheses over denominator A r e a open parentheses triangle C A B close parentheses end fraction equals fraction numerator C E squared over denominator C A squared end fraction rightwards double arrow fraction numerator space A r e a open parentheses triangle C E F close parentheses over denominator A r e a open parentheses triangle C A B close parentheses end fraction equals open parentheses fraction numerator C E over denominator C A end fraction close parentheses squared rightwards double arrow 1 half equals open parentheses fraction numerator C E over denominator C A end fraction close parentheses squared space space space open square brackets because f r o m space e q u a t i o n space left parenthesis 1 right parenthesis close square brackets rightwards double arrow fraction numerator C E over denominator C A end fraction equals square root of 1 half end root rightwards double arrow fraction numerator C E over denominator C A end fraction equals fraction numerator 1 over denominator square root of 2 end fraction rightwards double arrow fraction numerator C A minus E A over denominator C A end fraction equals fraction numerator 1 over denominator square root of 2 end fraction rightwards double arrow 1 minus fraction numerator E A over denominator C A end fraction equals fraction numerator 1 over denominator square root of 2 end fraction rightwards double arrow 1 minus fraction numerator 1 over denominator square root of 2 end fraction equals fraction numerator E A over denominator C A end fraction rightwards double arrow fraction numerator E A over denominator C A end fraction equals 1 minus fraction numerator 1 over denominator square root of 2 end fraction rightwards double arrow fraction numerator E A over denominator C A end fraction equals fraction numerator square root of 2 minus 1 over denominator square root of 2 end fraction

Answered by Vimala Ramamurthy | 2nd Oct, 2015, 03:47: PM