If, in the following figure, PA = 8 cm, PD = 4 cm, CD equal to 3 cm, then AB is :

Let ABCD is a rhombus in which AB = BC = CD = DA = diagonal CA.
In triangle ABC ,
AB = BC = CA , therefore ?B = 60°
?D = ?B = 60°
?A=180 - ?B = 180° - 60° =120°
?C = ?A=120°
Hence ,The ratio of diagonals = ?2 : 1

In a quadrilateral ABCD,
Given :- ?B = 90°
AB² + BC² + CD² = AC² + CD² = AD² { ? In triangle ABC , AB² + BC² = AC² }
So, in triangle ACD, angle opposite to AD = 90° ( by Converse of Pythagoras theorem)
? ?ACD = 90°

Given :- ?A = x° , ?B = y° , ?C = ( y + 20 )° and 4x ? y = 10
We know that , ?A + ?B + ?C = 180°
? x + y + ( y + 20 ) = 180°
? x + 2y = 180 - 20 = 160° ........... ( ? )
4x ? y = 10 ........... ( ? )
From ( ? ) and ( ? ) , we get
? y = 70 , x = 20
? The angles of the triangle are 20° , 70° , 90° .
Hence , the triangle will be Right-angle triangle .

From given figure , we can see that
Let AB = 16 m , OM = 6 m , AM = BM = 8 m and OA = r
In triangle AMO ,
By pythagoras theorem ,
OA² = AM² + OM²
OA² = 8² + 6² = 64 + 36
OA² = 100 ? OA = ? 100 = 10
? OA = r = 10 m
The radius of the circular lawn = 10 m