Test: Introduction To Heron's Formula


20 Questions MCQ Test Mathematics (Maths) Class 9 | Test: Introduction To Heron's Formula


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QUESTION: 1

The area of a triangle is 150 cm2 and its sides are in the ratio 3 : 4 : 5. What is its perimeter?

Solution:

Let coefficient of ratios be X
a = 3x
b = 4x
c = 5x

Using Heron's Formula,
s = (3+4+5)x/2 = 12x/2 = 6x



Now,
a = 5*3 = 15 cm
b = 5*4 = 20 cm
c = 5*5 = 25 cm

Perimeter = 15+20+25 = 60 cm

QUESTION: 2

The area of triangle, whose sides are 15 cm, 25 cm and 14 cm:

Solution:
QUESTION: 3

The area of an equilateral triangle of side 6 cm is​

Solution:
QUESTION: 4

The perimeter of a rhombus is 146 cm. One of its diagonals is 55 cm. The length of the other diagonal and area of the rhombus are​

Solution:

QUESTION: 5

The sides of a triangle are in the ratio of 3 : 4 : 5. If its perimeter is 36 cm, then what is its area?

Solution:

 

⇒ AB + BC + CA = 36 cm

⇒ 3x + 4x + 5x = 36 cm

⇒ 12x = 36

⇒ x = 3

∴ AB = 3x = 9

BC = 4x = 12

CA = 5x = 15

Now, (AB)2 + (BC)2

=  (9)2 + (12)2

=  81 + 144

=  255

= (AC)2

Δ ABC is a right angle triangle and ∠B is right angle.

Area of Δ ABC = 1/2 (AB x BC)

= 1/2 (9 x12)

= 54 cm2

 

 

QUESTION: 6

If each side of an equilateral triangle is increased by 2 cm, then its area increases by 3√3 cm2. The length of its each side and its area are respectively equal to

Solution:
QUESTION: 7

The area of quadrilateral PQRS, in which PQ = 7 cm, QR = 6 cm, RS = 12 cm, PS = 15 cm and PR = 9 cm:​

Solution:
QUESTION: 8

The base of an isosceles triangle is 10 cm and one of its equal sides is 13 cm. The area of the triangle is

Solution:
QUESTION: 9

The lengths of a triangle are 6 cm, 8 cm and 10 cm. Then the length of perpendicular from the opposite vertex to the side whose length is 8cm is:

Solution:
QUESTION: 10

The perimeter of an isosceles right angled triangle having an area of 200 cm2 is

Solution:

 Area of isosceles right angled triangle =  1/2 *b*h

Here if it is isosceles right angled triangle ,

Then perpendicular = base = x

So, area =1/2 *x*x

  400 = x²

  x² = 400

  x =  = 20 cm

  so   P = B = 20

  By Pythagoras theorem ,

  H² = P² +B²

  = (20)²+(20)²

  = 400 + 400

  = 800

H = 20 root 2 cm

Perimeter = 20 + 20 + 20 root 2

  = 40+20 root 2

  = 68.28 cm

QUESTION: 11

A carpenter had to make a triangle with sides 5, 6, 5 units. By mistake he made one with sides 5, 8, 5 units. The difference between their areas is:

Solution:

Area of triangle that carpenter had to made = (5 + 6 + 5)/2 {By Heron's formula}

= 8 sq. units

Area of triangle that carpenter made by mistake = ( 5 + 8 + 5)/2

= 9 sq. units

Hence, the difference between their areas is 1 sq. units

QUESTION: 12

The area of a right triangle of height 15 m and base 20 m is​

Solution:
QUESTION: 13

The sides of a triangle are in the ratio of 3 : 4 : 5. If its perimeter is 36 cm, then what is its area?

Solution:
QUESTION: 14

The area of an equilateral triangle of side 14 cm is

Solution:
QUESTION: 15

The sides of a triangle are in the ratio 25 : 17 : 12 and its perimeter is 540m. The area of the triangle is​

Solution:

The perimeter of a triangular field = 540m

Let the sides are 25x , 17x , 12 x

Perimeter of a ∆ = sum of three sides

25x + 17x + 12x = 540

 54x = 540

 x = 10 

1st side (a) - 25x = 25×10= 250m 

2nd side(b)= 17x = 17×10= 170m

3rd side (c)= 12x = 12 × 10 =120m

Semi - perimeter ( S) = a+b+c/2 

= (250 + 170+120)/2 = 540/2 = 270 m

 Area of the ∆= √ S(S - a)(S - b)(S - c)

[By Heron’s Formula]

= √ S(S - 250)(S - 170)(S - 120) 

= √ 270(270 - 250)(270 - 170)(270 - 120) 

= √ 270× 20×100×150 

= √ 81000000 

Area of the ∆= 9000 m²

Hence, the Area of the ∆= 9000 m²

QUESTION: 16

The area of an equilateral triangle with perimeter 18x is:

Solution:
QUESTION: 17

The sides of a triangular plot are in the ratio of 3 : 5 : 7 and its perimeter is 300 m. Its area is

Solution:

 Suppose that the sides, in metres, are 3x, 5x and 7x 

Then, we know that 3x + 5x + 7x = 300 (perimeter of the triangle) 

Therefore, 15x = 300, which gives x = 20. 

So the sides of the triangle are 3 × 20 m, 5 × 20 m and 7 × 20 m 

i.e., 60 m, 100 m and 140 


= 1500 √(3) = 2598.07621135

QUESTION: 18

The area of a right triangle with base 5 m and altitude 12 m is​

Solution:
QUESTION: 19

There is a slide in a park. One of its side walls has been painted in some colour with a message “Keep the park green and Clean”. If sides of the wall are 15m, 11 m and 6 m, the area painted in colour is________.

 

Solution:

QUESTION: 20

The area of a triangle whose sides are 13 cm, 14 cm and 15 cm is

Solution:

By using herons formula 
We get s=13+14+15/2
S=21cm
Now area=√s(s-a) (s-b) (s-c) 
√21(21-13)(21-14)(21-15)
√21(8)(7)(6)
√21(336)
√7056
84 cm² is the area.

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