8.MG.4a-e The student will apply the Pythagorean Theorem to solve problems involving right triangles, including those in context.
Understanding the Standard
The Pythagorean Theorem is essential for solving problems involving right triangles.
The hypotenuse of a right triangle is the side opposite the right angle.
The hypotenuse of a right triangle is always the longest side of the right triangle.
The legs of a right triangle form the right angle.
In a right triangle, the square of the length of the hypotenuse equals the sum of the squares of the legs. This relationship is known as the Pythagorean Theorem: a2+b2=c2.
The Pythagorean Theorem is used to determine the measure of any one of the three sides of a right triangle when the measures of the other two sides are known.
The converse of the Pythagorean Theorem states that if the square of the length of the hypotenuse equals the sum of the squares of the legs in a triangle, then the triangle is a right triangle. This can be used to determine whether a triangle is a right triangle given the measures of its three sides.
The triangle inequality theorem states that the sum of any two sides of a triangle is greater than or equal to the third side. This theorem can be explored to check for reasonableness of solutions.
Whole number triples that are the measures of the sides of right triangles, such as (3, 4, 5) and (5, 12, 13), are commonly known as Pythagorean triples. Additional sets of Pythagorean triples can be found by applying properties for similar triangles and proportional sides. For example, doubling the sides of a triangle with sides of (3, 4, 5) creates a Pythagorean triple of (6, 8, 10).
Introduction to the Pythagorean Theorem (00:00 - 01:45)
- Discussion of the legacy of the Pythagorean Theorem and its central formula: a^2 + b^2 = c^2.
- Identification of the different parts of a right triangle, specifically the legs and the hypotenuse.
Anatomy of a Right Triangle (01:45 - 03:15)
- Detailed look at the relationship between the sides, stressing that the hypotenuse is always the longest side and is located opposite the right angle.
- Explanation of how to properly assign values to the variables in the formula to ensure accurate calculations.
The Converse and Verification (03:15 - 04:30)
- Exploration of the Converse of the Pythagorean Theorem as a method to prove if a triangle is truly a right triangle.
- Using the formula as a 'verification check' for sets of measurements.
Pythagorean Triples and Proportionality (04:30 - 05:45)
- Introduction to common whole-number triples such as (3, 4, 5) and (5, 12, 13).
- Discussion on how similar triangles are formed by multiplying triples by a constant, such as doubling (3, 4, 5) to get (6, 8, 10).
The Triangle Inequality Theorem and Reasonableness (05:45 - 07:00)
- Overview of the Triangle Inequality Theorem, which states that the sum of any two sides must be greater than the third side.
- Importance of checking solutions for reasonableness, such as ensuring the hypotenuse remains the longest side in word problems and real-world contexts.