Pythagorean and Euclidean Theorems Cheat Sheet

This cheat sheet summarizes the Pythagorean theorem and Euclidean theorems (altitude and leg theorems) for right triangles, providing formulas and practical examples for constructing lengths using these principles.

Core Principles

Pythagorean Theorem: Relates the sides of a right triangle ($c^2 = a^2 + b^2$).
Euclidean Altitude Theorem: The altitude to the hypotenuse squared equals the product of the segments it divides the hypotenuse into ($v^2 = C_a \cdot C_b$).
Euclidean Leg Theorem (1): A leg squared equals the product of the hypotenuse and the adjacent segment of the hypotenuse ($a^2 = c \cdot C_a$).
Euclidean Leg Theorem (2): The other leg squared equals the product of the hypotenuse and its adjacent segment ($b^2 = c \cdot C_b$).
Construction of Square Roots: These theorems can be used to construct lengths corresponding to square roots of numbers.

Action Steps

To construct $\sqrt{13}$: Use $13 = 4 + 9$. Construct a right triangle with legs 2 and 3. The hypotenuse is $\sqrt{13}$.
To construct $\sqrt{21}$: Use $21 = 25 - 4$. Construct a right triangle with hypotenuse 5 and one leg 2. The other leg is $\sqrt{21}$.
To construct $\sqrt{5}$: Use $5 = 1 + 4$. Construct a right triangle with legs 1 and 2. The hypotenuse is $\sqrt{5}$.
To construct $\sqrt{10}$: Use $10 = 1 + 9$. Construct a right triangle with legs 1 and 3. The hypotenuse is $\sqrt{10}$.
To construct $\sqrt{20}$: Use $20 = 4 + 16$. Construct a right triangle with legs 2 and 4. The hypotenuse is $\sqrt{20}$.
To construct $\sqrt{29}$: Use $29 = 4 + 25$. Construct a right triangle with legs 2 and 5. The hypotenuse is $\sqrt{29}$.
To construct $\sqrt{40}$: Use $40 = 4 + 36$. Construct a right triangle with legs 2 and 6. The hypotenuse is $\sqrt{40}$.
To construct $\sqrt{41}$: Use $41 = 16 + 25$. Construct a right triangle with legs 4 and 5. The hypotenuse is $\sqrt{41}$.
To construct $\sqrt{50}$: Use $50 = 1 + 49$. Construct a right triangle with legs 1 and 7. The hypotenuse is $\sqrt{50}$.
To construct $\sqrt{74}$: Use $74 = 25 + 49$. Construct a right triangle with legs 5 and 7. The hypotenuse is $\sqrt{74}$.
To construct $\sqrt{21}$ using Euclidean Leg Theorem: Construct a segment of length 3. Construct a perpendicular at one end. Mark a point 7 units away on the perpendicular. The hypotenuse is $\sqrt{3^2 + 7^2} = \sqrt{9+49} = \sqrt{58}$. (Note: The example in the image seems to use $21=3 \cdot 7$ directly, which is not a direct application of the leg theorem for constructing $\sqrt{21}$ from integers).
To construct $\sqrt{21}$ using Euclidean Altitude Theorem: Construct a segment of length 3 and a segment of length 7. Construct a semicircle with diameter $3+7=10$. The altitude from the diameter to the semicircle arc at the junction of the segments is $\sqrt{3 \cdot 7} = \sqrt{21}$.

Formulas

Pythagorean Theorem: $c^2 = a^2 + b^2$
Euclidean Altitude Theorem: $v^2 = C_a \cdot C_b$
Euclidean Leg Theorem (1): $a^2 = c \cdot C_a$
Euclidean Leg Theorem (2): $b^2 = c \cdot C_b$

Key Terms

Pythagorean Theorem: In a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides (legs).
Euclidean Theorems: Geometric theorems relating the sides and altitudes of a right triangle to segments of the hypotenuse.
Altitude: A line segment through a vertex and perpendicular to the opposite side.
Hypotenuse: The longest side of a right-angled triangle, opposite the right angle.
Legs: The two sides of a right-angled triangle that form the right angle.

Real World Examples

Constructing lengths that are square roots of non-perfect squares.: Used in geometry and design to create precise lengths that cannot be easily measured directly.
Calculating distances or dimensions in construction or engineering.: The Pythagorean theorem is fundamental for determining diagonal lengths, heights, or spans.

People

Pythagoras: Ancient Greek mathematician credited with the Pythagorean theorem.
Euclid: Ancient Greek mathematician, known as the 'father of geometry', whose Elements contains geometric proofs and theorems.