# 29.  In the adjoining figure, ABCD is a rectangle.  Its diagonal AC =  15 cm and $\angle$ACD = $\alpha$.  If cot $\alpha =\frac{3}{2}$ , find the perimeter and area of the rectangle. 30.  Using the measurements given in figure alongside, (a) find the values of : (1) sin $\varphi$ (ii) tan $\theta$. (b) write an expression for AD in terms of $\theta$. Hint: (b) CD =  5. Draw DE perpendicular to AB, BE = 5, EA = 9.     Area of a rectangle = length*breadth

• 4
Solution to 29

cot(x) = base/perpendicular
CD = (3/2)x  # assume AD to be x

15^2 = ( (3/2)x)^2 + x^2 =
225 = (9/4)x^2 + x^2
225 = (13/4)x^2
x^2  = 225*(4/13)
x = 30/sqrt(13)

= (3/2) * 30/sqrt(13) = 45/ sqrt(13)

Perimeter is 2( AD + CD) = 2(30/sqrt(13) + 45/sqrt(13) ) = 150/sqrt(13)

Area is AD*CD = 30/sqrt(13) * 45/sqrt(13) = 1350/13 sq cm
• -5
30
DC2= DB2 - BC2
= 132 - 122 =  169 - 144 = 25
DC = 5 cm

If we draw DE perpendicular to AB such that E is on AB
BCDE is a rectangle
BC =DE = 12 cm
CD = BE = 5 cm
so AE = AB - BE = 14 - 5 = 9 cm

in triangle ADE ( right angled at D)
AE = 9 cm DE = 12 cm
AD^2 = AE^2 + DE^2 = 9^2 + 12^2 = 81 + 144 = 225
sin(EAD) = DE/DA = 12/15 = 4/5  #sin = perpendicular/hypotenuse
tan(EAD) = DE/EA = 12/9 = 4/3 # tan = perpendicular/base
• 6
DC be 3k where ki is positive real no