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The side of the equilateral triangle are shortened by 12 unit 13 unit and 14 unit respectively and a right angle triangle is formed the side of the equilateral triangle?
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Equilateral Triangle and Right Angle Triangle Relationship
An equilateral triangle has all three sides of equal length. Let the side of the equilateral triangle be x units.
Given that the sides of the equilateral triangle are shortened by 12 units, 13 units, and 14 units respectively to form a right-angle triangle.
Formation of Right-Angle Triangle
When one side of an equilateral triangle is shortened by 12 units, the new side length becomes (x-12) units.
When another side is shortened by 13 units, the new side length becomes (x-13) units.
When the third side is shortened by 14 units, the new side length becomes (x-14) units.
Right-Angle Triangle Formation
For a right-angle triangle to be formed, the Pythagorean theorem must be satisfied.
According to the Pythagorean theorem, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides.
Therefore, for the right-angle triangle to be formed:
(x-12)^2 + (x-13)^2 = (x-14)^2
Solving the Equation
Expanding and simplifying the equation leads to:
x^2 - 24x + 144 + x^2 - 26x + 169 = x^2 - 28x + 196
2x^2 - 50x + 313 = x^2 - 28x + 196
x^2 - 22x - 117 = 0
Calculating the Side of the Equilateral Triangle
Using the quadratic formula, we can solve for x:
x = [22 ± √(22^2 + 4*117)] / 2
x = [22 ± √(484 + 468)] / 2
x = [22 ± √952] / 2
x = [22 ± 2√238] / 2
x = 11 ± √238
Therefore, the side of the equilateral triangle is 11 + √238 units.
An equilateral triangle has all three sides of equal length. Let the side of the equilateral triangle be x units.
Given that the sides of the equilateral triangle are shortened by 12 units, 13 units, and 14 units respectively to form a right-angle triangle.
Formation of Right-Angle Triangle
When one side of an equilateral triangle is shortened by 12 units, the new side length becomes (x-12) units.
When another side is shortened by 13 units, the new side length becomes (x-13) units.
When the third side is shortened by 14 units, the new side length becomes (x-14) units.
Right-Angle Triangle Formation
For a right-angle triangle to be formed, the Pythagorean theorem must be satisfied.
According to the Pythagorean theorem, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides.
Therefore, for the right-angle triangle to be formed:
(x-12)^2 + (x-13)^2 = (x-14)^2
Solving the Equation
Expanding and simplifying the equation leads to:
x^2 - 24x + 144 + x^2 - 26x + 169 = x^2 - 28x + 196
2x^2 - 50x + 313 = x^2 - 28x + 196
x^2 - 22x - 117 = 0
Calculating the Side of the Equilateral Triangle
Using the quadratic formula, we can solve for x:
x = [22 ± √(22^2 + 4*117)] / 2
x = [22 ± √(484 + 468)] / 2
x = [22 ± √952] / 2
x = [22 ± 2√238] / 2
x = 11 ± √238
Therefore, the side of the equilateral triangle is 11 + √238 units.
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The side of the equilateral triangle are shortened by 12 unit 13 unit and 14 unit respectively and a right angle triangle is formed the side of the equilateral triangle?
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The side of the equilateral triangle are shortened by 12 unit 13 unit and 14 unit respectively and a right angle triangle is formed the side of the equilateral triangle? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about The side of the equilateral triangle are shortened by 12 unit 13 unit and 14 unit respectively and a right angle triangle is formed the side of the equilateral triangle? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The side of the equilateral triangle are shortened by 12 unit 13 unit and 14 unit respectively and a right angle triangle is formed the side of the equilateral triangle?.
The side of the equilateral triangle are shortened by 12 unit 13 unit and 14 unit respectively and a right angle triangle is formed the side of the equilateral triangle? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about The side of the equilateral triangle are shortened by 12 unit 13 unit and 14 unit respectively and a right angle triangle is formed the side of the equilateral triangle? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The side of the equilateral triangle are shortened by 12 unit 13 unit and 14 unit respectively and a right angle triangle is formed the side of the equilateral triangle?.
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