Pythagorean theorem Proofs Practice

Answer 1

The Pythagorean theorem 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. Mathematically, this is expressed as:

\[ c^2 = a^2 + b^2 \]

where \( c \) is the length of the hypotenuse, and \( a \) and \( b \) are the lengths of the other two sides. Below are several proofs of the Pythagorean theorem.

1. Geometric Proof (Square on the Hypotenuse)

1. Construct a Square: Draw a square with a side length of \( (a + b) \).
2. Vertices of the Square: Label the vertices A, B, C, D such that AB is horizontal.
3. Inside the Square: Now, inside this square, create a right triangle with legs \( a \) and \( b \), which we will replicate in each corner of the square.
4. Area Calculation:
   • The area of the large square is \( (a + b)^2 \).
   • The area conserved inside the square after fitting the triangles is \( c^2 \) (the area of the square created by using the hypotenuse of the triangles).
   • The area of the 4 triangles is \( 4 \cdot \frac{1}{2}ab = 2ab \).
5. Setting up the equation: \[ (a + b)^2 = c^2 + 2ab \] Expanding \( (a + b)^2 \) gives: \[ a^2 + 2ab + b^2 \] Therefore, setting the two equations equal gives: \[ a^2 + b^2 + 2ab = c^2 + 2ab \] Subtracting \( 2ab \) from both sides results in: \[ a^2 + b^2 = c^2 \]

2. Algebraic Proof (Using Similar Triangles)

1. Consider a Right Triangle: Let triangle ABC be a right triangle where \( \angle C = 90^\circ \).
2. Drop a Perpendicular: Drop a perpendicular from point C to the hypotenuse AB, intersecting at point D.
3. Create Two Similar Triangles: Triangle ADC and triangle BDC are similar to triangle ABC since they all have a right angle and share angles A and B, respectively.
4. Set Up Ratios: By the property of similar triangles: \[ \frac{AD}{AC} = \frac{AC}{AB} \quad \text{and} \quad \frac{BD}{BC} = \frac{BC}{AB} \] Let \( AD = x, BD = y, AC = b, BC = a, \) and \( AB = c \). Thus, we have:
   • \( \frac{x}{b} = \frac{b}{c} \Rightarrow x \cdot c = b^2 \Rightarrow x = \frac{b^2}{c} \)
   • \( \frac{y}{a} = \frac{a}{c} \Rightarrow y \cdot c = a^2 \Rightarrow y = \frac{a^2}{c} \)
5. Sum the Segments of hypotenuse: \[ c = x + y = \frac{b^2}{c} + \frac{a^2}{c} \]
6. Multiply through by c: \[ c^2 = b^2 + a^2 \]

3. Coordinate Proof

1. Place triangle in the coordinate system:
   • Let point A be at (0, 0), point B at (a, 0), and point C at (a, b).
2. Distance Formula: The distance formula provides a way to calculate lengths: \[ AB = a, \quad BC = b, \quad AC = c \]
3. Calculate lengths using distance formula:
   • \( AC = \sqrt{(a - 0)^2 + (b - 0)^2} = \sqrt{a^2 + b^2} \)
4. Square both sides: \[ AC^2 = a^2 + b^2 \Rightarrow c^2 = a^2 + b^2 \]

These proofs reflect the various methods used to establish the Pythagorean theorem. Each proof demonstrates that for every right triangle, the relationship defined by the theorem holds true. You can now try proving the theorem using the ideas above or even explore other proofs, such as those involving dissection or the use of algebraic manipulation!