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According to Pythagoras' Theorem, what is the relationship between the sides of a right-angled triangle?
- a)The sum of the lengths of the two shorter sides equals the length of the hypotenuse.
- b)The product of the lengths of the two shorter sides equals the length of the hypotenuse.
- c)The length of the hypotenuse squared equals the sum of the squares of the lengths of the other two sides.
- d)The length of the hypotenuse squared equals the product of the lengths of the other two sides.
Correct answer is option 'C'. Can you explain this answer?
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According to Pythagoras Theorem, what is the relationship between the ...
Pythagoras Theorem:
The Pythagoras Theorem states that in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
Explanation:
To understand the relationship between the sides of a right-angled triangle according to Pythagoras Theorem, let's consider a right-angled triangle ABC, where side AB is the hypotenuse and sides BC and AC are the other two sides.
Formula:
The Pythagoras Theorem can be mathematically represented as:
AB² = BC² + AC²
Proof:
1. Draw a right-angled triangle ABC.
2. Construct a square on each side of the triangle.
3. Let the side lengths of square ABDE be AB.
4. Let the side lengths of square BCGF be BC.
5. Let the side lengths of square CAHI be AC.
6. Now, arrange these squares as shown in the diagram below, where the vertices of the squares coincide with the vertices of the triangle.
Pythagorean Theorem Diagram:
```
A I
| |
| |
| |
| |
B_________C
/ /
/ /
/ /
/ /
D__________E
G H
```
Explanation of the Diagram:
- The square on side AB (ABDE) is placed on the opposite side of the triangle.
- The square on side BC (BCGF) is placed adjacent to side BC.
- The square on side AC (CAHI) is placed adjacent to side AC.
Analysis:
1. The area of square ABDE is equal to the sum of the areas of squares BCGF and CAHI, plus twice the area of triangle ABC.
- AB² = BC² + AC² + 2(0.5 * BC * AC)
2. The area of square ABDE is equal to the area of square BCGF plus the area of square CAHI.
- AB² = BC² + AC²
Conclusion:
From the above analysis and equations, we can conclude that the square of the length of the hypotenuse AB is equal to the sum of the squares of the lengths of the other two sides BC and AC. Therefore, option 'C' is correct: "The length of the hypotenuse squared equals the sum of the squares of the lengths of the other two sides."
The Pythagoras Theorem states that in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
Explanation:
To understand the relationship between the sides of a right-angled triangle according to Pythagoras Theorem, let's consider a right-angled triangle ABC, where side AB is the hypotenuse and sides BC and AC are the other two sides.
Formula:
The Pythagoras Theorem can be mathematically represented as:
AB² = BC² + AC²
Proof:
1. Draw a right-angled triangle ABC.
2. Construct a square on each side of the triangle.
3. Let the side lengths of square ABDE be AB.
4. Let the side lengths of square BCGF be BC.
5. Let the side lengths of square CAHI be AC.
6. Now, arrange these squares as shown in the diagram below, where the vertices of the squares coincide with the vertices of the triangle.
Pythagorean Theorem Diagram:
```
A I
| |
| |
| |
| |
B_________C
/ /
/ /
/ /
/ /
D__________E
G H
```
Explanation of the Diagram:
- The square on side AB (ABDE) is placed on the opposite side of the triangle.
- The square on side BC (BCGF) is placed adjacent to side BC.
- The square on side AC (CAHI) is placed adjacent to side AC.
Analysis:
1. The area of square ABDE is equal to the sum of the areas of squares BCGF and CAHI, plus twice the area of triangle ABC.
- AB² = BC² + AC² + 2(0.5 * BC * AC)
2. The area of square ABDE is equal to the area of square BCGF plus the area of square CAHI.
- AB² = BC² + AC²
Conclusion:
From the above analysis and equations, we can conclude that the square of the length of the hypotenuse AB is equal to the sum of the squares of the lengths of the other two sides BC and AC. Therefore, option 'C' is correct: "The length of the hypotenuse squared equals the sum of the squares of the lengths of the other two sides."
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According to Pythagoras Theorem, what is the relationship between the ...
An ancient Babylonian tablet displaying a concept predating Pythagoras by over a millennium suggests the Pythagorean theorem could be the world's oldest known case of plagiarism.
About Pythagoras:
- Pythagoras was a Greek philosopher and mathematician.
- Pythagoras lived during the 6th century B.C. on the island of Samos, Greece.
- He is commonly said to be the first pure mathematician who proposed that everything is a number.
- He is best known in the modern day for the Pythagorean Theorem.
- Pythagoras’ Theorem:
- It states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
- This formula has been applied to measuring distance and space as, for example, in planning and executing the construction of a building.
- Pythagorean Brotherhood:
- Pythagoras founded a secretive religious and philosophical community known as the Pythagorean Brotherhood.
- The members, known as Pythagoreans, followed a strict set of rules and beliefs, including the idea that mathematics and numbers held mystical and spiritual significance.
- It was here he taught that "the whole cosmos is a scale and a number.
Hence option c is the correct answer.
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According to Pythagoras Theorem, what is the relationship between the sides of a right-angled triangle?a)The sum of the lengths of the two shorter sides equals the length of the hypotenuse.b)The product of the lengths of the two shorter sides equals the length of the hypotenuse.c)The length of the hypotenuse squared equals the sum of the squares of the lengths of the other two sides.d)The length of the hypotenuse squared equals the product of the lengths of the other two sides.Correct answer is option 'C'. Can you explain this answer?
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