Danloader Closed WhatsApp Group
Here are the explanations for √3, √5, and √7:
√3:
*Geometric Method:*
Imagine an equilateral triangle with side length 2 units. Draw an altitude from one vertex to the opposite side, creating two 30-60-90 right triangles.
Using the Pythagorean theorem:
a^2 + b^2 = c^2
where a and b are the legs, and c is the hypotenuse.
1^2 + (√3)^2 = 2^2
1 + 3 = 4
√3 = √3
Now, imagine a hexagon with side length 1 unit. The diagonal of this hexagon is equal to √3 units.
Using a compass and straightedge, we can construct a sequence of triangles and hexagons to approximate the value of √3.
*Decimal Expansion:*
We can express √3 as a decimal using the following method:
√3 = 1 + 0.7 + 0.04 + 0.002 + ... (using the Babylonian method or other numerical methods)
This decimal expansion converges to approximately 1.7321.
√5:
*Geometric Method:*
Imagine a golden rectangle with length 2 units and width 1 unit. Draw a diagonal, creating two similar triangles.
Using the Pythagorean theorem:
a^2 + b^2 = c^2
where a and b are the legs, and c is the hypotenuse.
1^2 + 2^2 = (√5)^2
1 + 4 = 5
√5 = √5
Now, imagine a decagon with side length 1 unit. The diagonal of this decagon is equal to √5 units.
Using a compass and straightedge, we can construct a sequence of triangles and decagons to approximate the value of √5.
*Decimal Expansion:*
We can express √5 as a decimal using the following method:
√5 = 2 + 0.2 + 0.01 + 0.0005 + ... (using the Babylonian method or other numerical methods)
This decimal expansion converges to approximately 2.2361.
√7:
*Geometric Method:*
Imagine a heptagon with side length 1 unit. Draw a diagonal, creating two similar triangles.
Using the Pythagorean theorem:
a^2 + b^2 = c^2
where a and b are the legs, and c is the hypotenuse.
1^2 + (√7)^2 = (√14)^2
1 + 7 = 14
√7 = √7
Now, imagine a 14-gon with side length 1 unit. The diagonal of this 14-gon is equal to √7 units.
Using a compass and straightedge, we can construct a sequence of triangles and 14-gons to approximate the value of √7.
*Decimal Expansion:*
We can express √7 as a decimal using the following method:
√7 = 2 + 0.6 + 0.03 + 0.0015 + ... (using the Babylonian method or other numerical methods)
This decimal expansion converges to approximately 2.6458.
Here are the approximate values:
- √3 ≈ 1.7321
- √5 ≈ 2.2361
- √7 ≈ 2.6458
√3:
*Geometric Method:*
Imagine an equilateral triangle with side length 2 units. Draw an altitude from one vertex to the opposite side, creating two 30-60-90 right triangles.
Using the Pythagorean theorem:
a^2 + b^2 = c^2
where a and b are the legs, and c is the hypotenuse.
1^2 + (√3)^2 = 2^2
1 + 3 = 4
√3 = √3
Now, imagine a hexagon with side length 1 unit. The diagonal of this hexagon is equal to √3 units.
Using a compass and straightedge, we can construct a sequence of triangles and hexagons to approximate the value of √3.
*Decimal Expansion:*
We can express √3 as a decimal using the following method:
√3 = 1 + 0.7 + 0.04 + 0.002 + ... (using the Babylonian method or other numerical methods)
This decimal expansion converges to approximately 1.7321.
√5:
*Geometric Method:*
Imagine a golden rectangle with length 2 units and width 1 unit. Draw a diagonal, creating two similar triangles.
Using the Pythagorean theorem:
a^2 + b^2 = c^2
where a and b are the legs, and c is the hypotenuse.
1^2 + 2^2 = (√5)^2
1 + 4 = 5
√5 = √5
Now, imagine a decagon with side length 1 unit. The diagonal of this decagon is equal to √5 units.
Using a compass and straightedge, we can construct a sequence of triangles and decagons to approximate the value of √5.
*Decimal Expansion:*
We can express √5 as a decimal using the following method:
√5 = 2 + 0.2 + 0.01 + 0.0005 + ... (using the Babylonian method or other numerical methods)
This decimal expansion converges to approximately 2.2361.
√7:
*Geometric Method:*
Imagine a heptagon with side length 1 unit. Draw a diagonal, creating two similar triangles.
Using the Pythagorean theorem:
a^2 + b^2 = c^2
where a and b are the legs, and c is the hypotenuse.
1^2 + (√7)^2 = (√14)^2
1 + 7 = 14
√7 = √7
Now, imagine a 14-gon with side length 1 unit. The diagonal of this 14-gon is equal to √7 units.
Using a compass and straightedge, we can construct a sequence of triangles and 14-gons to approximate the value of √7.
*Decimal Expansion:*
We can express √7 as a decimal using the following method:
√7 = 2 + 0.6 + 0.03 + 0.0015 + ... (using the Babylonian method or other numerical methods)
This decimal expansion converges to approximately 2.6458.
Here are the approximate values:
- √3 ≈ 1.7321
- √5 ≈ 2.2361
- √7 ≈ 2.6458
Here are the explanations for √3, √5, and √7:
√3:
*Geometric Method:*
Imagine an equilateral triangle with side length 2 units. Draw an altitude from one vertex to the opposite side, creating two 30-60-90 right triangles.
Using the Pythagorean theorem:
a^2 + b^2 = c^2
where a and b are the legs, and c is the hypotenuse.
1^2 + (√3)^2 = 2^2
1 + 3 = 4
√3 = √3
Now, imagine a hexagon with side length 1 unit. The diagonal of this hexagon is equal to √3 units.
Using a compass and straightedge, we can construct a sequence of triangles and hexagons to approximate the value of √3.
*Decimal Expansion:*
We can express √3 as a decimal using the following method:
√3 = 1 + 0.7 + 0.04 + 0.002 + ... (using the Babylonian method or other numerical methods)
This decimal expansion converges to approximately 1.7321.
√5:
*Geometric Method:*
Imagine a golden rectangle with length 2 units and width 1 unit. Draw a diagonal, creating two similar triangles.
Using the Pythagorean theorem:
a^2 + b^2 = c^2
where a and b are the legs, and c is the hypotenuse.
1^2 + 2^2 = (√5)^2
1 + 4 = 5
√5 = √5
Now, imagine a decagon with side length 1 unit. The diagonal of this decagon is equal to √5 units.
Using a compass and straightedge, we can construct a sequence of triangles and decagons to approximate the value of √5.
*Decimal Expansion:*
We can express √5 as a decimal using the following method:
√5 = 2 + 0.2 + 0.01 + 0.0005 + ... (using the Babylonian method or other numerical methods)
This decimal expansion converges to approximately 2.2361.
√7:
*Geometric Method:*
Imagine a heptagon with side length 1 unit. Draw a diagonal, creating two similar triangles.
Using the Pythagorean theorem:
a^2 + b^2 = c^2
where a and b are the legs, and c is the hypotenuse.
1^2 + (√7)^2 = (√14)^2
1 + 7 = 14
√7 = √7
Now, imagine a 14-gon with side length 1 unit. The diagonal of this 14-gon is equal to √7 units.
Using a compass and straightedge, we can construct a sequence of triangles and 14-gons to approximate the value of √7.
*Decimal Expansion:*
We can express √7 as a decimal using the following method:
√7 = 2 + 0.6 + 0.03 + 0.0015 + ... (using the Babylonian method or other numerical methods)
This decimal expansion converges to approximately 2.6458.
Here are the approximate values:
- √3 ≈ 1.7321
- √5 ≈ 2.2361
- √7 ≈ 2.6458
0 Comments
·0 Shares
·0 Reviews