# Free Heron's Formula 03 Practice Test - 9th Grade

A triangular wall having sides 3 m, 5 m and 6 m has to be painted. If the cost of painting 1 m2 of a wall is ₹ 5, what is the cost of painting the whole wall?

A.

₹ 1613

B.

₹ 1014

C.

₹ 1410

D.

14

#### SOLUTION

Solution : B

Semi perimeter of a triangle = (a+b+c)2 (3+5+6)2=7m

Area of a triangle = s(sa)(sb)(sc)   7(73)(75)(76)

7×4×2×1
14×4

56=214 m2

Since,cost of painting 1 m2 of wall is ₹5,

The cost of painting the whole wall =5×214= 1014.

If the area of a rhombus-shaped box is 67cm2 and its one diagonal is 6cm. Find the side of the rhombus.

A.

8 cm

B.

3 cm

C.

4 cm

D.

5 cm

#### SOLUTION

Solution : C

Let the side of the rhombus = a

Area of the triangle containing the diagonal = 12 times area of the rhombus

One of the diagonal = 6 cm

Semi perimeter = 6+a+a2=a+3

Now the area of the triangle containing the diagonal =s(sa)(sb)(sc)

37 = (a+3)(a+3a)(a+3a)(a+36)
=  (a+3)(a3)(9)
= (a29)(9)
We take square on both sides and get,

63 = (a29)×9
(a29) = 7
a2 = 7+9 = 16

a= 4cm

If the side of a rhombus is 6 cm and its one diagonal is 8 cm. Find the area of the rhombus in cm2.

A.

93

B.

39

C.

165

D.

85

#### SOLUTION

Solution : C

Side of the rhombus = 6 cm

One of the diagonal = 8 cm

Semi perimeter of the triangle containing the diagonal = (6+6+8)2 = 10 cm

Area of the triangle containing the diagonal = s(sa)(sb)(sc)

= 10(106)(106)(108)

= 10×4×4×2
= 2×5×4×4×2
85cm2

Area of rhombus = 2 (Area of the triangle containing the diagonal)

Area of the rhombus = 2×85 = 165 cm2

The area of a rectangle is 96 cm2 and its length is 12 cm. The perimeter of the rectangle is same as the perimeter of a triangle. Find the semi perimeter of the triangle.

A.

10 cm

B.

40 cm

C.

20 cm

D.

80 cm

#### SOLUTION

Solution : C

Area of a rectangle = length × breadth

96 = 12 × breadth

Breadth = 8 cm

Perimeter of the rectangle = 2(length + breadth) = 2(12+8) cm = 40 cm

Perimeter of the rectangle = Perimeter of the triangle

Therefore, semi perimeter of the triangle =402=20 cm

State whether true or false:
​If in a right angled triangle only two sides are given, then its area can be calculated.

A.

True

B.

False

#### SOLUTION

Solution : A

In a right angled triangle, if the base and the height are given, the area can be calculated directly by using 12 × base × height. If the hypotenuse and either the base or the height are given, then the third side can be found using Pythagoras' theorem. Then the area can be calculated using the same formula.

The sides of a triangular plot are in the ratio of 16: 8: 10 and its perimeter is 340 m. Its area in m2 is 100A. The value of A is

___

#### SOLUTION

Solution :

Let the sides of the triangle be 16x, 8x and 10x.

Then, 16x + 8x + 10x = 340
34x = 340
x = 34034
x =10.

Hence the sides are 160 cm, 80 cm and 100 cm.
Using the Heron's formula, Area A = s(sa)(sb)(sc)

Semi-Perimeter s = perimeter2
=3402
=170cm

A=170(170160)(170100)(17080)
=170×10×70×90
=1001071cm2

The value of A is 1071

The area of an equilateral triangle whose side is 2 cm is A. The integral value of A is___

#### SOLUTION

Solution :

The area of an equilateral triangle is 34a2 , where a = 2 cm.
Hence the area is 34×22
=3 cm2

Thus the value of A = 3.

The sides of a triangle are 5 cm, 6 cm and 3 cm long. The area of the triangle is A cm2. Then the value of A is ___.

#### SOLUTION

Solution : s=5+6+32
=7 cm
Area A=s(sa)(sb)(sc)
=7(75)(76)(73)
=7×2×1×4
=56 cm2
The value of A is 56.

Which of the following are correct regarding the given figure? A.

The area of a parallelogram is 120 cm2

B.

The area of a parallelogram is 180 cm2

C.

The length of the altitude is 15 cm

D.

The length of the altitude is 30 cm

#### SOLUTION

Solution : B and C

For BCD,
s=25+17+122=27

Using Heron's formula,
A=27(2725)(2717)(2712)
=27×2×10×15
=3×3×3×2×2×5×3×5
=3×3×2×5
=90 cm2

The parallelogram can also be divided into 2 triangles by diagonal AC, as shown below, the area of each triangle being =90 cm2.

Area of parallelogram =90×2=180 cm2 In, ACD
Area of ACD=12×base×Altitude
90=12×12×Altitude
Altitude=15 cm

The perimeter of a triangular field is 420 m and its sides are in the ratio 6:7:8. Find the area of the triangular field.

A. 42015 m2
B. 10507 m2
C. 105015 m2
D. 210015 m2

#### SOLUTION

Solution : D

Let the sides be 6x, 7x, 8x.
Then, 6x+7x+8x=420
21x=420
x=20
the sides are:
a =120 m, b =140 m, c =160 m
Semi-perimeter s = 4202=210 m
A=s(sa)(sb)(sc)
=210(210120)(210140)(210160)
=210×90×70×50
=210015 m2