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Understanding the Pythagorean Theorem

Introduction to Pythagoras and Right Triangles

• Daniel Carrión introduces the topic of the Pythagorean theorem, emphasizing its significance in mathematics.

• He explains that a right triangle has one angle measuring 90 degrees, with the longest side called the hypotenuse (denoted as 'c').

• The other two sides are referred to as catheti, represented by 'a' and 'b'.

Basics of Exponents

• Carrión discusses squaring numbers, explaining that raising a number to the second power means multiplying it by itself.

• Examples include:

• 3^2 = 3 times 3

• 2^2 = 2 times 2

The Pythagorean Theorem Explained

• The theorem states that in any right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides: c^2 = a^2 + b^2.

• A visual representation is provided using a right triangle with sides measuring 3 cm and 4 cm.

Application of the Theorem

• Carrión substitutes values into the formula:

• For hypotenuse c = 5, cathetus a = 3, and cathetus b = 4.

• Calculation shows:

• 5^2 = 25

• 3^2 + 4^2 = 9 + 16 = 25, confirming that it satisfies the theorem.

Further Examples

• Another example involves finding an unknown hypotenuse where catheti measure:

• a = 6,cm, b = 5,cm.

• Using Pythagorean theorem:

• Set up equation: c^2 = a^2 + b^2 Rightarrow c^2 = (6)^2 + (5)^2 Rightarrow c^2 =36 +25 Rightarrow c^2 =61.

Finding Unknown Sides

• To find hypotenuse, take square root:

• sqrt61 ≈7.81,cm.

Example with Known Hypotenuse

• In another case, given:

• Hypotenuse (c) is 10,cm,

• Cathetus (b) is 8,cm,,

we need to find cathetus (a).

Rearranging for Unknown Side

• Rearranging gives us:

• Formula becomes:

[c² − b²=a².]

Final Calculation Steps

• Substitute known values into rearranged formula:

• Calculate:

[10²−8²=a² ⇒100−64=a² ⇒36=a².]

Conclusion on Cathetus Measurement

• Taking square root yields:

Understanding the Pythagorean Theorem

Introduction to the Problem

• The problem involves two legs of a right triangle, denoted as catetos a and b. The hypotenuse measures 15 centimeters, while cateto a is 12 centimeters. The goal is to find the length of cateto b using the Pythagorean theorem.

Rearranging the Formula

• To solve for b, we rearrange the Pythagorean theorem formula c^2 = a^2 + b^2. By isolating b^2, we rewrite it as b^2 = c^2 - a^2.

Substituting Values

• We substitute known values into our rearranged formula: c = 15 (the hypotenuse) and a = 12. This gives us b^2 = 15^2 - 12^2.

Calculating Values

• Performing the calculations:

• Calculate 15^2 = 225

• Calculate 12^2 = 144

• Thus, b^2 = 225 - 144 = 81.

• Taking the square root yields b = sqrt81 = 9, indicating that cateto b measures 9 centimeters.

Conclusion and Further Exercises