Q1) PQRS is a rectangle in which PQ = 9 cm and PS = 6 cm. 
T is a point on PQ such that PT = 7 cm and RV is
the perpendicular from R to ST. Calculate ST and
RV. 

I managed to calculate ST but not RV.
                       _____    _____    __
I assume you got ST = Ö6²+7² = Ö36+49 = Ö85

By alternate interior angles, ÐRSV = ÐPTS
    
Right triangles RVS and SPT are similar, since two 
triangles are similar if 2 angles of one triangle 
have the same measures as 2 angles of the other,
and we know that both contain a right angle.

Therefore

 RV     RS 
———— = ————
 PS     ST

 RV      9 
———— = —————
 6      Ö85
     __
 RV×Ö85 = 9×6
     __
 RV×Ö85 = 54
          __
 RV = 54/Ö85 cm exactly or about 5.86 cm

Q2) In a pararllelogram ABCD, the diagonal AC is at
right angles to AB. If AB = 12 cm and BC = 13 cm, 
find the area of the parallelogram.

D_____C
 \   |\
  \  | \13
   \ |  \
    \|___\
    A  12 B

That figure is not to scale, as it should be leaning 
farther to the left, but it will do for our purpose.

Area of parallelogram = base × height =

                            AB × AC = 12 × AC

We find AC by the Pythagorean theorem
      _________    _________    __
AC = Ö13² + 12² = Ö169 - 144 = Ö25 = 5

So area of parallelogram = 12 × AC = 12 × 5 = 60 cm²

Edwin
AnlytcPhil@aol.com