All questions of Heron's Formula for Class 9 Exam

Given:
Base of the triangle = 8 cm
Height of the triangle = 10 cm

To find:
Area of the triangle

Formula:
Area of the triangle = 1/2 × base × height

Solution:
Substituting the given values in the formula, we get
Area of the triangle = 1/2 × 8 cm × 10 cm
Area of the triangle = 40 cm²

Therefore, the correct option is (d) 40 cm².

Use heron's formula

30 m ^2...

Given : isosceles right angle triangle
therefore base = height
area of triangle = 8 cm ^2
to find : Hypotenuse
procedure : consider base=height=X

area of right triangle = 1/2×base× height = 8
= 1/2×X×X =8
= X^2/2 = 8
= X^2=16
=X=4

Hypotenuse^2=base^2+height ^2
H^2=4^2+4^2
H^2=16+16
H^2=32
therefore. H = 32^1/2

Solution:

Given,
Base of right triangle = 8 cm
Hypotenuse of right triangle = 10 cm

Let the height of the right triangle be h.

Using Pythagoras theorem, we have

(hypotenuse)2 = (base)2 + (height)2
102 = 82 + h2
100 = 64 + h2
h2 = 36
h = 6 cm

Therefore, the area of the right triangle is given by

Area = 1/2 x base x height
= 1/2 x 8 x 6
= 24 cm2

Hence, the correct option is (a) 24cm2.

Side=14cm
area of equilateral triangle=(√3/4)×a^2 where a=side
using this formula,
(√3/4) × 14 ×14=49√3 cm^2

The correct answer is b) 9 cm².

To find the area of an equilateral triangle, we can use the formula:

Area = (sqrt(3) / 4) * side^2

Given that the side of the equilateral triangle is 6 cm, we can substitute this value into the formula:

Area = (sqrt(3) / 4) * 6^2
= (sqrt(3) / 4) * 36
= (1.732 / 4) * 36
= 0.433 * 36
= 15.588

Therefore, the area of the equilateral triangle is approximately 15.588 cm², which rounds to 16 cm².

To find the cost of painting the triangular board, we need to find the area of the triangle first.

**Finding the Area of the Triangle:**

We can use Heron's formula to find the area of the triangle when the lengths of its sides are known. Heron's formula states that the area of a triangle with sides a, b, and c is given by:

Area = √(s(s-a)(s-b)(s-c))

where s is the semi-perimeter of the triangle and is calculated as:

s = (a + b + c) / 2

In this case, the lengths of the sides are given as 6 cm, 8 cm, and 10 cm. So, we can substitute these values into the formulas to find the area.

**Calculating the Semi-Perimeter:**

s = (6 + 8 + 10) / 2
= 24 / 2
= 12 cm

**Calculating the Area:**

Area = √(12(12-6)(12-8)(12-10))
= √(12 * 6 * 4 * 2)
= √(576)
= 24 cm²

So, the area of the triangular board is 24 cm².

**Calculating the Cost of Painting:**

The cost of painting is given as 9 paise per cm². To find the total cost, we need to multiply the area of the board by the cost per cm².

Cost = Area * Cost per cm²
= 24 cm² * 9 paise/cm²
= 216 paise

Since 1 rupee is equal to 100 paise, we can convert the cost to rupees by dividing by 100.

Cost = 216 paise / 100
= Rs. 2.16

Therefore, the cost of painting the triangular board is Rs. 2.16, which corresponds to option B.

Using Heron's formula, we can calculate the area of the triangle:

s = (5 + 8 + 11)/2 = 12

A = √(s(s-5)(s-8)(s-11))
A = √(12(12-5)(12-8)(12-11))
A = √(12(7)(4)(1))
A = √(336)
A ≈ 18.33 m²

Therefore, the area of the flower bed is approximately 18.33 square meters.

Given information:
Angle A = 30°
Side b = 4 units
Side c = 6 units

Step 1: Find side a using the Law of Cosines
Using the Law of Cosines formula:
a² = b² + c² - 2bc * cos(A)
a² = 4² + 6² - 2*4*6 * cos(30°)
a² = 16 + 36 - 48 * cos(30°)
a² = 52 - 48 * 0.866 (cos(30°) is approximately 0.866)
a² = 52 - 41.568
a² = 10.432
a ≈ √10.432
a ≈ 3.23 units

Step 2: Calculate the area of the triangle using the formula
Area of triangle ABC = 0.5 * b * c * sin(A)
Area = 0.5 * 4 * 6 * sin(30°)
Area = 0.5 * 24 * 0.5
Area = 6 cm²
Therefore, the correct answer is option b) 6 cm².

Let 2 equal sides of isoceles right angled triangle be = x
Given,
Area = 8 cm2
so, 8=1/2×x×x
x=√16=4
Now , we have got the the other two sides of the triangle so let's apply Pythagoras theorem to get the length of hypotenuse
Let hypotenuse be = H
4^2+4^2=H^2
16+16=H^2
H=√32=4√2

16√3 cm. square is correct

Given:
Area of rhombus = 96 cm²
One diagonal = 16 cm

To find:
Length of a side of the rhombus

Formula used:
Area of rhombus = (d₁×d₂)/2, where d₁ and d₂ are the diagonals of the rhombus

Explanation:
1. We know that the area of a rhombus is given by the formula:
Area = (d₁×d₂)/2, where d₁ and d₂ are the diagonals of the rhombus.
Substituting the given values, we get:
96 = (16×d₂)/2
Simplifying, we get:
d₂ = (96×2)/16 = 12 cm

2. Now, we know that the diagonals of a rhombus are perpendicular bisectors of each other.
Hence, we can use the Pythagorean theorem to find the length of a side of the rhombus.

3. Let ABCD be a rhombus with diagonals AC and BD intersecting at O. Let OD = 8 cm (half of diagonal BD) and OB = x cm (one side of rhombus). Then, we have:

AO = OC = 8 cm (since diagonals bisect each other)
AB = CD = x (since opposite sides of a rhombus are equal)

Using Pythagoras theorem in triangle AOB, we get:
AB² = AO² + OB²
x² = 8² + x²
Simplifying, we get:
x² - x² = 64
2x² = 64
x² = 32
x = √32 = 4√2

Hence, the length of a side of the rhombus is 4√2 cm.

Therefore, the correct option is (d) 10 cm.

Ar one triangle =125cm^2
therefore ar four triangles =125cm^2×4=500cm^2
(since all 4 triangles of rhombus have equal area)

The area of an equilateral triangle can be found using the formula:

Area = (sqrt(3) / 4) * (side)^2

Plugging in the side length of 4 cm:

Area = (sqrt(3) / 4) * (4)^2
Area = (sqrt(3) / 4) * 16
Area = (1.732 / 4) * 16
Area = 0.433 * 16
Area = 6.928 cm^2

So the correct answer is 6.928 cm^2, not 8 cm^2.

To find the area of an equilateral triangle, we can use the formula:

Area = (√3/4) * side^2

Given that each side of the equilateral triangle measures 10 cm, we can substitute this value into the formula:

Area = (√3/4) * 10^2

Simplifying the equation further:

Area = (√3/4) * 100

Calculating the value of (√3/4):

(√3/4) ≈ 0.433

Substituting this value back into the equation:

Area ≈ 0.433 * 100

Area ≈ 43.3 cm^2

Therefore, the correct answer is option 'A', 43.3 cm^2.

Given:
Ratio of sides of the triangular plot = 3:5:7
Perimeter of the triangular plot = 300 m

To find:
Area of the triangular plot

Let the sides of the triangle be 3x, 5x, and 7x.

Perimeter of the triangle = sum of all three sides
3x + 5x + 7x = 300
15x = 300
x = 20

Therefore, the sides of the triangle are 3x = 60 m, 5x = 100 m, and 7x = 140 m.

Area of a triangle can be calculated using Heron's formula:
Area = √(s(s-a)(s-b)(s-c))
where s is the semi-perimeter of the triangle, and a, b, and c are the lengths of its sides.

Let's calculate the semi-perimeter:
s = (60 + 100 + 140)/2
s = 300/2
s = 150

Now, we can calculate the area using Heron's formula:
Area = √(150(150-60)(150-100)(150-140))
Area = √(150*90*50*10)
Area = √(67500000)
Area ≈ 2598.08 m²

Therefore, the area of the triangular plot is approximately 2598.08 m², which corresponds to option A.

Edge of triangle are = 6cm, 8 cm and 10cm
S = a + b + C/2
S = 6+8+10/2
S = 24/2
S=12
area is = √s (S-a)(S-b) (S-C)
area is =√12 (12-6) (12-8) (12-10)
area is =√12x6x4x2
area is = √2x6x6X2X2X2
area is = 6x2x2 cm cm²
Area is = 24cm²
We convert Paise into Rupee
1 paise = 1/100 rupee
70 Paise = 70/100 rupee
70 paise = 0.7 rupee
The cost of Painting 24cm²=24x 0.7cm²
The cost of Painting = 16.80 cm²(Ans.)