SUMMARY: This chapter introduces Heron's Formula for calculating the area of a triangle when the lengths of all three sides are known. KEY TOPICS: Heron's Formula, semi-perimeter, area of a triangle, derivation of Heron's Formula, application of Heron's Formula, solving problems using Heron's Formula, examples with integer side lengths, examples with decimal side lengths, comparison with other area formulas.
What is the semi-perimeter of a triangle with sides 7 cm, 8 cm, and 9 cm?
A12 cm
B13 cm
C14 cm
D15 cm
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Correct answer: Option 2 — 13 cm
Q21 Mark
Using Heron's Formula, what is the area of a triangle with sides 10 cm, 10 cm, and 12 cm?
A48 cm²
B50 cm²
C52 cm²
D60 cm²
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Correct answer: Option 1 — 48 cm²
Q31 Mark
If the area of a triangle is 30 cm² and the lengths of two sides are 5 cm and 7 cm, what is the length of the third side using Heron's Formula?
A8 cm
B9 cm
C10 cm
D11 cm
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Correct answer: Option 3 — 10 cm
Q41 Mark
Which of the following is the correct formula for the area of a triangle using Heron's Formula?
AArea = √(s(s-a)(s-b)(s-c))
BArea = (a + b + c) / 2
CArea = (1/2) * base * height
DArea = a * b * c
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Correct answer: Option 1 — Area = √(s(s-a)(s-b)(s-c))
Q51 Mark
A triangle has sides of lengths 6.5 cm, 7.5 cm, and 8.5 cm. What is the area of the triangle using Heron's Formula?
A18.5 cm²
B19.5 cm²
C20.5 cm²
D21.5 cm²
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Correct answer: Option 2 — 19.5 cm²
Short Answer Questions5 questions
Q63 Marks
What is Heron's Formula and how is it used to calculate the area of a triangle?
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Heron's Formula states that the area of a triangle can be calculated using the formula A = √(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 the sides. It is particularly useful when the lengths of all three sides are known.
Q73 Marks
Define semi-perimeter and explain how it is calculated for a triangle with sides of lengths 5 cm, 6 cm, and 7 cm.
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The semi-perimeter 's' of a triangle is defined as half of the sum of its side lengths. For a triangle with sides 5 cm, 6 cm, and 7 cm, the semi-perimeter is calculated as s = (5 + 6 + 7) / 2 = 9 cm.
Q83 Marks
Using Heron's Formula, calculate the area of a triangle with sides measuring 8 cm, 15 cm, and 17 cm.
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First, calculate the semi-perimeter: s = (8 + 15 + 17) / 2 = 20 cm. Then, apply Heron's Formula: A = √(20(20-8)(20-15)(20-17)) = √(20 × 12 × 5 × 3) = √(3600) = 60 cm².
Q93 Marks
What is the significance of Heron's Formula compared to other area formulas for triangles?
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Heron's Formula is significant because it allows for the calculation of the area of a triangle when only the lengths of the sides are known, without needing to know the height. This is particularly useful in cases where height is difficult to measure or not provided, unlike the base-height formula.
Q103 Marks
Explain how Heron's Formula can be applied to triangles with decimal side lengths, and provide an example with sides 4.5 cm, 5.5 cm, and 6.5 cm.
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Heron's Formula can be applied to triangles with decimal side lengths in the same way as with integer lengths. For sides 4.5 cm, 5.5 cm, and 6.5 cm, first calculate the semi-perimeter: s = (4.5 + 5.5 + 6.5) / 2 = 8.5 cm. Then, use the formula: A = √(8.5(8.5-4.5)(8.5-5.5)(8.5-6.5)) = √(8.5 × 4 × 3 × 2) = √(204) ≈ 14.28 cm².
Long Answer Questions5 questions
Q116 Marks
Using Heron's Formula, calculate the area of a triangle with side lengths 7 cm, 8 cm, and 9 cm. Show all steps including the calculation of the semi-perimeter.
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To find the area of the triangle, first calculate the semi-perimeter (s) using the formula s = (a + b + c) / 2. Here, a = 7 cm, b = 8 cm, and c = 9 cm, so s = (7 + 8 + 9) / 2 = 12 cm. Then, apply Heron's Formula: Area = √[s(s-a)(s-b)(s-c)] = √[12(12-7)(12-8)(12-9)] = √[12 × 5 × 4 × 3] = √720 = 26.83 cm² (approximately).
Q126 Marks
Derive Heron's Formula starting from the basic formula for the area of a triangle. Explain each step in the derivation process.
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Heron's Formula can be derived from the standard area formula A = 1/2 × base × height. By dropping a perpendicular from a vertex to the base, we can express the height in terms of the sides of the triangle. Let the sides be a, b, and c, and the semi-perimeter s = (a + b + c) / 2. Using the Pythagorean theorem, we can express the area in terms of s and the sides. After manipulating the equations and substituting the height, we arrive at Heron's Formula: Area = √[s(s-a)(s-b)(s-c)].
Q136 Marks
A triangle has sides of lengths 10.5 m, 12.3 m, and 14.7 m. Calculate the area using Heron's Formula and discuss the significance of using this formula for non-integer side lengths.
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First, calculate the semi-perimeter: s = (10.5 + 12.3 + 14.7) / 2 = 18.75 m. Then, use Heron's Formula: Area = √[s(s-a)(s-b)(s-c)] = √[18.75(18.75-10.5)(18.75-12.3)(18.75-14.7)] = √[18.75 × 8.25 × 6.45 × 4.05] = √[4860.78] = 69.8 m² (approximately). Heron's Formula is particularly useful for triangles with non-integer side lengths as it allows for precise area calculations without needing to determine height.
Q146 Marks
Compare Heron's Formula with the traditional formula for the area of a triangle (1/2 × base × height). Discuss the advantages of using Heron's Formula in specific scenarios.
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Heron's Formula provides a way to calculate the area of a triangle when the height is not readily available, which is often the case in practical situations. Unlike the traditional formula, which requires knowledge of the height, Heron's Formula only requires the lengths of all three sides. This is particularly advantageous in cases where the triangle is irregular or when measurements are taken in the field. Additionally, Heron's Formula can be applied to any triangle, making it a versatile tool in geometry.
Q156 Marks
A triangle has sides measuring 5 cm, 12 cm, and 13 cm. Verify if it is a right triangle using Heron's Formula and calculate its area.
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To verify if the triangle is a right triangle, check if the square of the longest side equals the sum of the squares of the other two sides: 13² = 5² + 12², which gives 169 = 25 + 144, confirming it is a right triangle. Now, calculate the area using Heron's Formula: semi-perimeter s = (5 + 12 + 13) / 2 = 15 cm. Area = √[s(s-a)(s-b)(s-c)] = √[15(15-5)(15-12)(15-13)] = √[15 × 10 × 3 × 2] = √[900] = 30 cm². Thus, the area is 30 cm².
Assertion–Reason Questions5 questions
Q161 Mark
Assertion (A): Heron's Formula can be used to find the area of a triangle when the lengths of all three sides are known.
Reason (R): The formula is derived from the semi-perimeter of the triangle.
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Correct answer: Option 1 —
Both A and R are true, and R is the correct explanation of A.
Q171 Mark
Assertion (A): The semi-perimeter of a triangle is half the sum of its side lengths.
Reason (R): The semi-perimeter is denoted by the letter 's' and is calculated as s = (a + b + c) / 2.
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Correct answer: Option 1 —
Both A and R are true, and R is the correct explanation of A.
Q181 Mark
Assertion (A): Heron's Formula is applicable only to triangles with integer side lengths.
Reason (R): Heron's Formula can be applied to triangles with both integer and decimal side lengths.
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Correct answer: Option 3 —
A is true, but R is false.
Q191 Mark
Assertion (A): The area of a triangle calculated using Heron's Formula is always greater than the area calculated using the base-height formula.
Reason (R): The area calculated using Heron's Formula can be less than, equal to, or greater than the area calculated using the base-height formula depending on the triangle's dimensions.
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Correct answer: Option 3 —
A is true, but R is false.
Q201 Mark
Assertion (A): If a triangle has sides of lengths 5, 12, and 13, its area can be calculated using Heron's Formula.
Reason (R): The triangle with these side lengths is a right triangle, and Heron's Formula can be used for any triangle.
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Correct answer: Option 1 —
Both A and R are true, and R is the correct explanation of A.
Statement-Based Questions5 questions
Q211 Mark
Statement 1: Heron's Formula can be used to find the area of a triangle when only the lengths of the sides are known.
Statement 2: The semi-perimeter of a triangle is calculated by adding the lengths of all three sides and dividing by three.
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Correct answer: Option 2 —
Only Statement 1 is true.
Q221 Mark
Statement 1: Heron's Formula is derived from the Pythagorean theorem.
Statement 2: The area of a triangle using Heron's Formula is calculated as √(s(s-a)(s-b)(s-c)) where s is the semi-perimeter.
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Correct answer: Option 3 —
Only Statement 2 is true.
Q231 Mark
Statement 1: Heron's Formula can only be applied to triangles with integer side lengths.
Statement 2: The semi-perimeter is half the sum of the lengths of the sides of the triangle.
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Correct answer: Option 4 —
Both statements are false.
Q241 Mark
Statement 1: The area of a triangle calculated using Heron's Formula can be compared with the area calculated using base and height.
Statement 2: Heron's Formula is applicable for any triangle, regardless of the type.
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Correct answer: Option 1 —
Both statements are true.
Q251 Mark
Statement 1: To find the semi-perimeter, you need to know the lengths of all three sides of the triangle.
Statement 2: Heron's Formula provides a way to calculate the area without knowing the height of the triangle.
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Correct answer: Option 1 —
Both statements are true.
