3) What is the formula used for the Pythagorean theorem?
4) What type of triangle does the Pythagorean theorem apply to?
5) Give at least 1 example on how the Pythagorean theorem works.
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The Pythagorean Theorem, also referred to as the ‘Pythagoras theorem,’ is arguably the most famous formula in mathematics that defines the relationships between the sides of a right triangle.
The theorem is attributed to a Greek mathematician and philosopher named Pythagoras (569-500 B.C.E.). He has many contributions to mathematics, but the Pythagorean Theorem is the most important of them.
Pythagoras is credited with several contributions in mathematics, astronomy, music, religion, philosophy, etc. One of his notable contributions to mathematics is the discovery of the Pythagorean Theorem. Pythagoras studied the sides of a right triangle and discovered that the sum of the square of the two shorter sides of the triangles is equal to the square of the longest side.
This article will discuss what the Pythagorean Theorem is, its converse, and the Pythagorean Theorem formula. Before getting deeper into the topic, let’s recall the right triangle. A right triangle is a triangle with one interior angle equals 90 degrees. In a right triangle, the two short legs meet at an angle of 90 degrees. The hypotenuse of a triangle is opposite the 90-degree angle.
What is the Pythagorean Theorem?
The Pythagoras theorem is a mathematical law that states that the sum of squares of the lengths of the two short sides of the right triangle is equal to the square of the length of the hypotenuse.
The Pythagoras theorem is algebraically written as:
a2 + b2 = c2
How to do the Pythagorean theorem?
Consider a right triangle above.
Given that:
∠ ABC= 90°.
Let BD be the perpendicular line to the side AC.
Similar ∆s:
∆ADB and ∆ABC are similar triangles.
From the similarity rule,
⇒ AD/AB = AB/AC
⇒ AD × AC = (AB) 2 —————– (i)
Similarly;
∆BDC and ∆ABC are similar triangles. Therefore;
⇒ DC/BC = BC/AC
⇒ DC × AC = (BC) 2 —————– (ii)
By combining equation (i) and (ii), we get,
AD × AC + DC × AC = (AB) 2 + (BC) 2
⇒ (AD + DC) × AC = (AB) 2 + (BC) 2
⇒ (AC)2 = (AB) 2 + (BC) 2
Therefore, if we let AC = c; AB = b and BC = b, then;
⇒ c2 = a2 + b2
There are many demonstrations of the Pythagorean Theorem given by different mathematicians.
Another common demonstration is to draw the 3 squares in such a way that they form a right triangle in between, and the area of the bigger square (the one at hypotenuse) is equal to the sum of the area of the smaller two squares (the ones on the two sides).
Consider the 3 squares below:
They are drawn in such a way that they form a right triangle. We can write their areas can in equation form:
Area of Square III = Area of Square I + Area of Square II
Let’s suppose the length of square I, square II, and square III are a, b and c, respectively.
Then,
Area of Square I = a 2
Area of Square II = b 2
Area of Square III = c 2
Hence, we can write it as:
a 2 + b 2 = c 2
which is a Pythagorean Theorem.
The Converse of the Pythagorean Theorem
The converse of the Pythagorean theorem is a rule that is used to classify triangles as either right triangle, acute triangle, or obtuse triangle.
Given the Pythagorean Theorem, a2 + b2 = c2, then:
For an acute triangle, c2< a2 + b2, where c is the side opposite the acute angle.
For a right triangle, c2= a2 + b2, where c is the side of the 90-degree angle.
For an obtuse triangle, c2> a2 + b2, where c is the side opposite the obtuse angle.
Example 1
Classify a triangle whose dimensions are; a = 5 m, b = 7 m and c = 9 m.
Solution
According to the Pythagorean Theorem, a2 + b2 = c2 then;
a2 + b2 = 52 + 72 = 25 + 49 = 74
But, c2 = 92 = 81
Compare: 81 > 74
Hence, c2 > a2 + b2 (obtuse triangle).
Example 2
Classify a triangle whose side lengths a, b, c, are 8 mm, 15 mm, and 17 mm, respectively.
Answers & Comments
Answer:
Pythagorean Theorem
Answer the following:
1) What is Pythagorean Theorem? -
2) Who discovered Pythagorean Theorem?
3) What is the formula used for the Pythagorean theorem?
4) What type of triangle does the Pythagorean theorem apply to?
5) Give at least 1 example on how the Pythagorean theorem works.
1
SEE ANSWER
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miyayeah1 avatar
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edcelanndelatorre
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6 answers
475 people helped
Answer:
The Pythagorean Theorem, also referred to as the ‘Pythagoras theorem,’ is arguably the most famous formula in mathematics that defines the relationships between the sides of a right triangle.
The theorem is attributed to a Greek mathematician and philosopher named Pythagoras (569-500 B.C.E.). He has many contributions to mathematics, but the Pythagorean Theorem is the most important of them.
Pythagoras is credited with several contributions in mathematics, astronomy, music, religion, philosophy, etc. One of his notable contributions to mathematics is the discovery of the Pythagorean Theorem. Pythagoras studied the sides of a right triangle and discovered that the sum of the square of the two shorter sides of the triangles is equal to the square of the longest side.
This article will discuss what the Pythagorean Theorem is, its converse, and the Pythagorean Theorem formula. Before getting deeper into the topic, let’s recall the right triangle. A right triangle is a triangle with one interior angle equals 90 degrees. In a right triangle, the two short legs meet at an angle of 90 degrees. The hypotenuse of a triangle is opposite the 90-degree angle.
What is the Pythagorean Theorem?
The Pythagoras theorem is a mathematical law that states that the sum of squares of the lengths of the two short sides of the right triangle is equal to the square of the length of the hypotenuse.
The Pythagoras theorem is algebraically written as:
a2 + b2 = c2
How to do the Pythagorean theorem?
Consider a right triangle above.
Given that:
∠ ABC= 90°.
Let BD be the perpendicular line to the side AC.
Similar ∆s:
∆ADB and ∆ABC are similar triangles.
From the similarity rule,
⇒ AD/AB = AB/AC
⇒ AD × AC = (AB) 2 —————– (i)
Similarly;
∆BDC and ∆ABC are similar triangles. Therefore;
⇒ DC/BC = BC/AC
⇒ DC × AC = (BC) 2 —————– (ii)
By combining equation (i) and (ii), we get,
AD × AC + DC × AC = (AB) 2 + (BC) 2
⇒ (AD + DC) × AC = (AB) 2 + (BC) 2
⇒ (AC)2 = (AB) 2 + (BC) 2
Therefore, if we let AC = c; AB = b and BC = b, then;
⇒ c2 = a2 + b2
There are many demonstrations of the Pythagorean Theorem given by different mathematicians.
Another common demonstration is to draw the 3 squares in such a way that they form a right triangle in between, and the area of the bigger square (the one at hypotenuse) is equal to the sum of the area of the smaller two squares (the ones on the two sides).
Consider the 3 squares below:
They are drawn in such a way that they form a right triangle. We can write their areas can in equation form:
Area of Square III = Area of Square I + Area of Square II
Let’s suppose the length of square I, square II, and square III are a, b and c, respectively.
Then,
Area of Square I = a 2
Area of Square II = b 2
Area of Square III = c 2
Hence, we can write it as:
a 2 + b 2 = c 2
which is a Pythagorean Theorem.
The Converse of the Pythagorean Theorem
The converse of the Pythagorean theorem is a rule that is used to classify triangles as either right triangle, acute triangle, or obtuse triangle.
Given the Pythagorean Theorem, a2 + b2 = c2, then:
For an acute triangle, c2< a2 + b2, where c is the side opposite the acute angle.
For a right triangle, c2= a2 + b2, where c is the side of the 90-degree angle.
For an obtuse triangle, c2> a2 + b2, where c is the side opposite the obtuse angle.
Example 1
Classify a triangle whose dimensions are; a = 5 m, b = 7 m and c = 9 m.
Solution
According to the Pythagorean Theorem, a2 + b2 = c2 then;
a2 + b2 = 52 + 72 = 25 + 49 = 74
But, c2 = 92 = 81
Compare: 81 > 74
Hence, c2 > a2 + b2 (obtuse triangle).
Example 2
Classify a triangle whose side lengths a, b, c, are 8 mm, 15 mm, and 17 mm, respectively.
Solution
a2 + b2 = 82 + 152 = 64 + 225 = 289
But, c2 = 172 = 289
Compare:289 = 289
Therefore, c2 = a2 + b2 (right triangle).
Example 3
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