Complex Problems:
1.If the area of a square with side a s equal to the area of a triangle with base a, then the altitude of the triangle is?
sol: area of a square with side a = a ² sq unts
area of a triangle with base a = ½ * a*h sq unts
a ² =1/2 *a *h => h = 2a
altitude of the triangle is 2a
2.An equilateral triangle is described on the diagonal of a square.
What is the ratio of the area of the triangle to that of the square?
Sol: area of a square = a ² sq cm
length of the diagonal = √2a cm
area of equilateral triangle with side √2a
= √3/4 * (√2a) ²
required ratio = √3a² : a ² = √3 : 2
3.The ratio of bases of two triangles is x:y and that of their areas is a:b. Then the ratio of their corresponding altitudes will be?
sol: a/b =(½ * x*H) /(1/2 * y * h)
bxH = ayh =>H/h =ay/bx
Hence H:h = ay:bx
4 .A parallelogram has sides 30m and 14m and one of its diagonals is 40m long. Then its area is?
sol: let ABCD be the given parallelogram
area of parallelogram ABCD = 2* (area of triangle ABC)
now a = 30m, b = 14m and c = 40m
s = ½(30+14+40) = 42m
Area of triangle ABC = √[ s(s-a)(s-b)(s-c) = √(42*12*28*2 = 168sq m
area of parallelogram ABCD = 2 *168 =336 sq m
5.If a parallelogram with area p, a triangle with area R and a triangle with area T are all constructed on the same base and all have the same altitude, then which of the following statements is false?
Sol: let each have base = b and height = h
then p = b*h, R = b*h and T = ½ * b*h
so P = R, P = 2T and T = ½ R are all correct statements
6.If the diagonals of a rhombus are 24cm and 10cm the area
and the perimeter of the rhombus are respectively.
Sol: area = ½*diagonal 1 *diagonal 2= ½ * 24 * 10= 120 sq cm
½ * diagonal 1 = ½ * 24 = 12cm
½ * diagonal 2 = ½ *10 =5 cm
side of a rhombus = (12) ² + (5) ² = 169 => AB = 13cm
7.If a square and a rhombus stand on the same base, then the ratio of the areas of the square and the rhombus is:
sol: A square and a rhombus on the same base are equal in area
8.The area of a field in the shape of a trapezium measures 1440sq m. The perpendicular distance between its parallel sides is 24cm.
If the ratio of the sides is 5:3, the length of the longer parallel side is:
sol: area of field =1/2 *(5x+3x) *24 = 96x sq m
96x = 1440 => x = 1440 /96 = 15
hence, the length of longer parallel side = 5x = 75m
9.The area of a circle of radius 5 is numerically what percent its circumference?
Sol: required percentage = π(5)²/(2π*5) *100 = 250%
10.A man runs round a circular field of radius 50m at the speed of 12m/hr. What is the time taken by the man to take twenty rounds of the field?
Sol: speed = 12 k/h = 12 * 5/18 = 10/3 m/s
distance covered = 20 * 2*22/7*50 = 44000/7m
time taken = distance /speed = 44000/7 * 3/10 = 220/7min
11.A cow s tethered in the middle of a field with a 14feet long rope. If the cow grazes 100 sq feet per day, then approximately what time will be taken by the cow to graze the whole field?
Sol: area of the field grazed = 22/7 * 14 * 14 = 616 sq feet
12.A wire can be bent in the form of a circle of radius 56cm.
If it is bent in the form of a square, then its area will be
sol: length of wire = 2π r = 2 *22/7 *56 = 352 cm
side of the square = 352/4 = 88cm
area of the square = 88*88 = 7744sq cm
13.The no of revolutions a wheel of diameter 40cm makes in traveling a distance of 176m is
sol: distance covered in 1 revolution = 2π r = 2 *22/7 *20 = 880/7 cm required no of revolutions = 17600 *7/880 = 140
14.The wheel of a motorcycle 70cm in diameter makes 40 revolutions in every 10sec.What is the speed of motorcycle n km/hr?
Sol: distance covered in 10sec = 2 *22/7 *35/100 *40 =88m
distance covered in 1 sec =88/10m = 8.8m
speed =8.8m/s = 8.8 * 18/5 *k/h = 31.68 k/h
