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HERON'S FORMULA - Q&A

Exercise 10.1

1. A traffic signal board, indicating 'SCHOOL AHEAD', is an equilateral triangle with side 'a'. Find the area of the signal board, using Heron's formula. If its perimeter is 180 cm, what will be the area of the signal board?

Step 1: Find the area of the equilateral triangle with side 'a' using Heron's formula.
Let the sides of the triangle be a, a, and a.
Semi-perimeter (s) = (a + a + a) / 2 = 3a/2.

Using Heron's formula, Area = √[s(s - a)(s - b)(s - c)]
Here, a = b = c = a.
Area = √[3a/2 (3a/2 - a) (3a/2 - a) (3a/2 - a)]
Area = √[3a/2 (a/2) (a/2) (a/2)]
Area = √[3a4 / 16]
Area = (√3 / 4) a2.

Step 2: Find the area when perimeter is 180 cm.
Perimeter = 180 cm.
Since it is an equilateral triangle, 3a = 180 cm.
Side a = 180 / 3 = 60 cm.

Substitute a = 60 in the area formula:
Area = (√3 / 4) (60)2
Area = (√3 / 4) × 3600
Area = 900√3 cm2.

2. The triangular side walls of a flyover have been used for advertisements. The sides of the walls are 122 m, 22 m and 120 m (see Fig. 10.6). The advertisements yield an earning of ₹ 5000 per m2 per year. A company hired one of its walls for 3 months. How much rent did it pay?

Step 1: Find the area of one triangular wall.
The sides are a = 122 m, b = 22 m, c = 120 m.
Semi-perimeter s = (122 + 22 + 120) / 2 = 264 / 2 = 132 m.

Using Heron's Formula:
Area = √[s(s - a)(s - b)(s - c)]
Area = √[132 (132 - 122) (132 - 22) (132 - 120)]
Area = √[132 × 10 × 110 × 12]
Area = √[(11 × 12) × 10 × (11 × 10) × 12]
Area = √[11 × 11 × 12 × 12 × 10 × 10]
Area = 11 × 12 × 10
Area = 1320 m2.

Step 2: Calculate the rent.
Rate = ₹ 5000 per m2 per year.
Rent for 1 year (12 months) for 1 m2 = ₹ 5000.
Rent for 3 months for 1 m2 = (5000 × 3) / 12 = ₹ 1250.

Total rent for 1320 m2 for 3 months:
Total Rent = 1320 × 1250
Total Rent = ₹ 16,50,000.

3. 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" (see Fig. 10.7). If the sides of the wall are 15 m, 11 m and 6 m, find the area painted in colour.

Step 1: Identify the sides of the triangle.
a = 15 m, b = 11 m, c = 6 m.

Step 2: Calculate semi-perimeter (s).
s = (15 + 11 + 6) / 2 = 32 / 2 = 16 m.

Step 3: Calculate the area using Heron's Formula.
Area = √[s(s - a)(s - b)(s - c)]
Area = √[16 (16 - 15) (16 - 11) (16 - 6)]
Area = √[16 × 1 × 5 × 10]
Area = √[16 × 50]
Area = √[16 × 25 × 2]
Area = 4 × 5 × √2
Area = 20√2 m2.

Answer: The area painted in colour is 20√2 m2.

4. Find the area of a triangle two sides of which are 18cm and 10cm and the perimeter is 42cm.

Step 1: Find the third side (c).
Perimeter = a + b + c
42 = 18 + 10 + c
42 = 28 + c
c = 42 - 28 = 14 cm.

Step 2: Calculate semi-perimeter (s).
s = Perimeter / 2 = 42 / 2 = 21 cm.

Step 3: Calculate area.
Area = √[s(s - a)(s - b)(s - c)]
Area = √[21 (21 - 18) (21 - 10) (21 - 14)]
Area = √[21 × 3 × 11 × 7]
Area = √[(3 × 7) × 3 × 11 × 7]
Area = √[3 × 3 × 7 × 7 × 11]
Area = 3 × 7 × √11
Area = 21√11 cm2.

5. Sides of a triangle are in the ratio of 12 : 17 : 25 and its perimeter is 540cm. Find its area.

Step 1: Find the actual lengths of the sides.
Let the sides be 12x, 17x, and 25x.
Perimeter = 12x + 17x + 25x = 540
54x = 540
x = 10.
So, the sides are:
a = 12 × 10 = 120 cm
b = 17 × 10 = 170 cm
c = 25 × 10 = 250 cm

Step 2: Calculate semi-perimeter (s).
s = 540 / 2 = 270 cm.

Step 3: Calculate area.
Area = √[s(s - a)(s - b)(s - c)]
Area = √[270 (270 - 120) (270 - 170) (270 - 250)]
Area = √[270 × 150 × 100 × 20]
Area = √[27 × 10 × 15 × 10 × 100 × 2 × 10]
Area = √[(9 × 3) × (3 × 5) × (2) × 100000]
Area = √[9 × 9 × 100 × 100 × 100]
Let's simplify differently:
Area = √[270 × 150 × 100 × 20]
Area = √[81,000,000]
Area = 9000 cm2.

6. An isosceles triangle has perimeter 30 cm and each of the equal sides is 12 cm. Find the area of the triangle.

Step 1: Find the third side.
Let equal sides a = 12 cm, b = 12 cm. Let third side be c.
Perimeter = a + b + c = 30
12 + 12 + c = 30
24 + c = 30
c = 6 cm.

Step 2: Calculate semi-perimeter (s).
s = 30 / 2 = 15 cm.

Step 3: Calculate area.
Area = √[s(s - a)(s - b)(s - c)]
Area = √[15 (15 - 12) (15 - 12) (15 - 6)]
Area = √[15 × 3 × 3 × 9]
Area = √[15 × 9 × 9]
Area = 9√15 cm2.

Answer: The area of the triangle is 9√15 cm2.

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Quick Review Flashcards - Click to flip and test your knowledge!
Question
What is the standard formula for the area of a triangle when the height is known?
Answer
The area is calculated as \frac{1}{2} \times \text{base} \times \text{height}.
Question
Under what condition is Heron's formula typically used to find the area of a triangle?
Answer
It is used when the lengths of all three sides are known but the height is not easily determined.
Question
According to the text, in which ancient city did Heron possibly live and work?
Answer
Heron worked in Alexandria, Egypt.
Question
Term: Heron of Alexandria
Answer
Definition: An encyclopaedic writer born around 10 C.E. who worked extensively in applied mathematics and mensuration.
Question
How many books comprise Heron's geometrical works on mensuration?
Answer
His geometrical works on mensuration are written in three books.
Question
Which specific book by Heron contains the derivation of the formula for the area of a triangle in terms of its three sides?
Answer
The derivation is found in Book I.
Question
Besides triangles, name three other shapes discussed in Heron's Book I.
Answer
Book I covers squares, rectangles, trapezoids (trapezia), regular polygons, circles, cylinders, cones, and spheres.
Question
What is an alternative name for Heron's formula mentioned in the text?
Answer
It is also known as Hero's formula.
Question
State Heron's formula for the area of a triangle.
Answer
The area is \sqrt{s(s-a)(s-b)(s-c)}.
Question
In Heron's formula, what do the variables a, b, and c represent?
Answer
These variables represent the lengths of the three sides of the triangle.
Question
In Heron's formula, what does the variable s represent?
Answer
The variable s represents the semi-perimeter of the triangle.
Question
Formula: Semi-perimeter (s)
Answer
s = \frac{a + b + c}{2}.
Question
Concept: Semi-perimeter
Answer
Definition: It is a value equal to half the perimeter of a polygon.
Question
If a triangular park has sides of 40\text{ m}, 24\text{ m}, and 32\text{ m}, what is its semi-perimeter (s)?
Answer
The semi-perimeter is 48\text{ m}.
Question
For a triangle with s = 48\text{ m} and sides a = 40\text{ m}, b = 24\text{ m}, and c = 32\text{ m}, calculate the value of (s - a).
Answer
The value of (s - a) is 8\text{ m}.
Question
For a triangle with s = 48\text{ m} and sides a = 40\text{ m}, b = 24\text{ m}, and c = 32\text{ m}, calculate the value of (s - b).
Answer
The value of (s - b) is 24\text{ m}.
Question
For a triangle with s = 48\text{ m} and sides a = 40\text{ m}, b = 24\text{ m}, and c = 32\text{ m}, calculate the value of (s - c).
Answer
The value of (s - c) is 16\text{ m}.
Question
What is the area of a triangular park with sides 40\text{ m}, 24\text{ m}, and 32\text{ m}?
Answer
The area is 384\text{ m}^2.
Question
How can the sides 32, 24, and 40 be used to verify if a triangle is right-angled?
Answer
By checking if the square of the longest side equals the sum of the squares of the other two sides (32^2 + 24^2 = 40^2).
Question
If the sides of a triangle are 32\text{ m}, 24\text{ m}, and 40\text{ m}, which side acts as the hypotenuse?
Answer
The side measuring 40\text{ m} is the hypotenuse.
Question
For an equilateral triangle with side length 10\text{ cm}, what is the value of the semi-perimeter s?
Answer
The semi-perimeter is 15\text{ cm}.
Question
Using Heron's formula, calculate the area of an equilateral triangle with a side of 10\text{ cm}.
Answer
The area is 25\sqrt{3}\text{ cm}^2.
Question
In an isosceles triangle with equal sides of 5\text{ cm} and an unequal side of 8\text{ cm}, what is the semi-perimeter s?
Answer
The semi-perimeter is 9\text{ cm}.
Question
What is the area of an isosceles triangle with sides 5\text{ cm}, 5\text{ cm}, and 8\text{ cm}?
Answer
The area is 12\text{ cm}^2.
Question
Two sides of a triangle are 8\text{ cm} and 11\text{ cm}, and the total perimeter is 32\text{ cm}. What is the length of the third side?
Answer
The third side is 13\text{ cm}.
Question
If the perimeter of a triangle is 32\text{ cm}, what is its semi-perimeter s?
Answer
The semi-perimeter is 16\text{ cm}.
Question
Calculate the area of a triangle with sides 8\text{ cm}, 11\text{ cm}, and 13\text{ cm}.
Answer
The area is 8\sqrt{30}\text{ cm}^2.
Question
A triangular park has sides 120\text{ m}, 80\text{ m}, and 50\text{ m}. What is the total perimeter?
Answer
The total perimeter is 250\text{ m}.
Question
A triangular park has sides 120\text{ m}, 80\text{ m}, and 50\text{ m}. What is the semi-perimeter s?
Answer
The semi-perimeter is 125\text{ m}.
Question
What is the area of a triangular park with side lengths 120\text{ m}, 80\text{ m}, and 50\text{ m}?
Answer
The area is 375\sqrt{15}\text{ m}^2.
Question
If a 3\text{ m} space is left for a gate in a fence surrounding a park with a 250\text{ m} perimeter, what is the total length of the fencing wire needed?
Answer
The length of wire needed is 247\text{ m}.
Question
How is the total cost of fencing calculated given a perimeter (minus gate) of 247\text{ m} and a rate of £20 per metre?
Answer
The cost is 247 \times 20, which equals £4940.
Question
If the sides of a triangular plot are in the ratio 3:5:7 and the perimeter is 300\text{ m}, how is the common factor x determined?
Answer
It is determined by solving the equation 3x + 5x + 7x = 300.
Question
In a triangle with sides in the ratio 3:5:7 and a perimeter of 300\text{ m}, what is the value of the common factor x?
Answer
The common factor x is 20.
Question
What are the specific side lengths of a triangle with sides in ratio 3:5:7 and a perimeter of 300\text{ m}?
Answer
The side lengths are 60\text{ m}, 100\text{ m}, and 140\text{ m}.
Question
What is the semi-perimeter s of a triangle with sides 60\text{ m}, 100\text{ m}, and 140\text{ m}?
Answer
The semi-perimeter is 150\text{ m}.
Question
What is the area of a triangular plot with sides 60\text{ m}, 100\text{ m}, and 140\text{ m}?
Answer
The area is 1500\sqrt{3}\text{ m}^2.
Question
A traffic signal board is an equilateral triangle with side 'a'. Express its semi-perimeter in terms of 'a'.
Answer
The semi-perimeter is \frac{3a}{2}.
Question
If a traffic signal board is an equilateral triangle with a perimeter of 180\text{ cm}, what is the length of each side?
Answer
Each side is 60\text{ cm}.
Question
The side walls of a flyover are 122\text{ m}, 22\text{ m}, and 120\text{ m}. What is the semi-perimeter s of these walls?
Answer
The semi-perimeter is 132\text{ m}.
Question
If the perimeter of a triangle is 42\text{ cm} and two sides are 18\text{ cm} and 10\text{ cm}, what is the length of the third side?
Answer
The third side is 14\text{ cm}.
Question
A triangle has sides in the ratio 12:17:25 and a perimeter of 540\text{ cm}. What is the value of the constant ratio x?
Answer
The constant ratio x is 10.
Question
What are the side lengths of a triangle with ratio 12:17:25 and perimeter 540\text{ cm}?
Answer
The side lengths are 120\text{ cm}, 170\text{ cm}, and 250\text{ cm}.
Question
An isosceles triangle has a perimeter of 30\text{ cm} and equal sides of 12\text{ cm}. What is the length of the unequal side?
Answer
The unequal side is 6\text{ cm}.
Question
The _____ of a triangle is defined as half of its perimeter.
Answer
semi-perimeter
Question
The formula \text{Area} = \sqrt{s(s-a)(s-b)(s-c)} is known as _____ formula.
Answer
Heron's
Question
In the context of Heron's formula, the term 2s is equivalent to the _____ of the triangle.
Answer
perimeter
Question
Heron's formula is particularly helpful for calculating the area of a _____ triangle when the height is unknown.
Answer
scalene
Question
Heron's Book I deals with problems on _____, covering areas of various polygons and surfaces of solids.
Answer
mensuration
Question
What is the semi-perimeter of a triangle with sides 15\text{ m}, 11\text{ m}, and 6\text{ m}?
Answer
The semi-perimeter is 16\text{ m}.
Question
If s = 16\text{ m} and the sides of a triangle are 15\text{ m}, 11\text{ m}, and 6\text{ m}, what is the calculated area?
Answer
The area is 20\sqrt{2}\text{ m}^2.
Question
How do you calculate the semi-perimeter s if you are given the perimeter P directly?
Answer
The semi-perimeter is calculated as s = \frac{P}{2}.
Question
If the sides of a triangle are 122\text{ m}, 22\text{ m}, and 120\text{ m}, find the area.
Answer
The area is 1320\text{ m}^2.