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Proof of Heron's Formula for the Area of an Isosceles Triangle with Equal Sides
Given: Isosceles triangle with two equal sides of length 'b'
Herons's Formula:
The area of a triangle with sides of lengths a, b, and c is given by:
\[Area = \sqrt{s(s-a)(s-b)(s-c)}\]
where \(s = \frac{a+b+c}{2}\) is the semi-perimeter of the triangle.
Proof:
Step 1: Determine the lengths of the sides
Since the triangle is isosceles with two equal sides of length 'b', the third side will also be 'b'.
Step 2: Calculate the semi-perimeter
\[s = \frac{b + b + b}{2} = \frac{3b}{2}\]
Step 3: Substitute the values into Heron's Formula
\[Area = \sqrt{\frac{3b}{2} \left(\frac{3b}{2} - b\right)^2}\]
\[Area = \sqrt{\frac{3b}{2} \left(\frac{b}{2}\right)^2}\]
\[Area = \sqrt{\frac{3b}{2} \cdot \frac{b^2}{4}}\]
\[Area = \sqrt{\frac{3b^3}{8}}\]
\[Area = \frac{b}{2} \sqrt{3}\]
Conclusion:
The area of an isosceles triangle with two equal sides of length 'b' using Heron's Formula is \(\frac{b}{2} \sqrt{3}\).
Given: Isosceles triangle with two equal sides of length 'b'
Herons's Formula:
The area of a triangle with sides of lengths a, b, and c is given by:
\[Area = \sqrt{s(s-a)(s-b)(s-c)}\]
where \(s = \frac{a+b+c}{2}\) is the semi-perimeter of the triangle.
Proof:
Step 1: Determine the lengths of the sides
Since the triangle is isosceles with two equal sides of length 'b', the third side will also be 'b'.
Step 2: Calculate the semi-perimeter
\[s = \frac{b + b + b}{2} = \frac{3b}{2}\]
Step 3: Substitute the values into Heron's Formula
\[Area = \sqrt{\frac{3b}{2} \left(\frac{3b}{2} - b\right)^2}\]
\[Area = \sqrt{\frac{3b}{2} \left(\frac{b}{2}\right)^2}\]
\[Area = \sqrt{\frac{3b}{2} \cdot \frac{b^2}{4}}\]
\[Area = \sqrt{\frac{3b^3}{8}}\]
\[Area = \frac{b}{2} \sqrt{3}\]
Conclusion:
The area of an isosceles triangle with two equal sides of length 'b' using Heron's Formula is \(\frac{b}{2} \sqrt{3}\).
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area of isosceles triangle using heron's formula proof.given equal sides is b
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area of isosceles triangle using heron's formula proof.given equal sides is b for Class 8 2024 is part of Class 8 preparation. The Question and answers have been prepared according to the Class 8 exam syllabus. Information about area of isosceles triangle using heron's formula proof.given equal sides is b covers all topics & solutions for Class 8 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for area of isosceles triangle using heron's formula proof.given equal sides is b.
area of isosceles triangle using heron's formula proof.given equal sides is b for Class 8 2024 is part of Class 8 preparation. The Question and answers have been prepared according to the Class 8 exam syllabus. Information about area of isosceles triangle using heron's formula proof.given equal sides is b covers all topics & solutions for Class 8 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for area of isosceles triangle using heron's formula proof.given equal sides is b.
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