What decimal number is illustrated?

Learning Objectives Apply the Pythagorean Theorem to find unknown side lengths of right triangles. Calculate the exact square root of non-perfect squares to determine irrational lengths. Estimate the decimal value of irrational numbers by identifying consecutive integers they lie between. Approximate irrational numbers to a specified decimal place using perfect squares or a calculator. Illustrate the approximate location of irrational numbers on a number line. Solve real-world problems involving irrational lengths derived from geometric illustrations. Ever wondered how we measure distances that aren't perfectly straight or don't land on a whole number? 📏 Get ready to discover how the Pythagorean Theorem helps us find these tricky lengths! In this lesson, you&#039...

TermDefinitionExample Pythagorean TheoremA fundamental relationship in Euclidean geometry among the three sides of a right triangle. It states that 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).If a right triangle has legs of length 3 and 4, the hypotenuse squared is 3² + 4² = 9 + 16 = 25. So, the hypotenuse is √25 = 5. HypotenuseThe longest side of a right triangle, which is always opposite the right (90-degree) angle.In a right triangle with sides 5 cm, 12 cm, and 13 cm, the 13 cm side is the hypotenuse. Legs (of a right triangle)The two shorter sides of a right triangle that form the right angle.In a right triangle with sides 5 cm, 12 cm, and 13 cm, the 5 cm and 12 cm sides are the legs. Square RootA nu...

Pythagorean Theorem $$a^2 + b^2 = c^2$$ Used to find the length of an unknown side in a right triangle when the lengths of the other two sides are known. 'a' and 'b' are the lengths of the legs, and 'c' is the length of the hypotenuse. Finding a Leg (Pythagorean Theorem variation) $$a = \sqrt{c^2 - b^2}$$ or $$b = \sqrt{c^2 - a^2}$$ Used to find the length of one leg when the hypotenuse and the other leg are known. Remember to subtract the square of the known leg from the square of the hypotenuse before taking the square root. Approximating Square Roots To approximate $\sqrt{x}$, find the two consecutive perfect squares that $x$ lies between. Then, the square root of $x$ will lie between the square roots of those perfect squares. This ru...