Mathematics
Grade 10
15 min
Converse of the Pythagorean theorem
Converse of the Pythagorean theorem
What you'll learn
- Identify the relationship between different customary units of length (inches, feet, yards, miles) with 100% accuracy.
- Solve word problems involving the conversion of customary units of length (inches, feet, yards, miles) with at least 80% accuracy.
- Explain the process of converting a larger customary unit of length (e.g., yards) to a smaller unit (e.g., inches) using multiplication, providing a clear and accurate explanation in at least 3 out of 4 attempts.
- Apply knowledge of customary length conversions to compare the lengths of two objects expressed in different units (e.g., comparing 3 feet to 40 inches) and correctly identify which is longer in 4 out of 5 trials.
Tutorial Preview
1
Introduction & Learning Objectives
Learning Objectives
State the Converse of the Pythagorean theorem in their own words.
Use the side lengths of a triangle to determine if it is a right triangle.
Differentiate between the Pythagorean theorem and its converse.
Apply the Pythagorean Inequalities to classify a triangle as acute, obtuse, or right.
First verify that three given side lengths can form a valid triangle using the Triangle Inequality Theorem.
Solve geometric and real-world problems by applying the converse of the Pythagorean theorem.
You're building a bookshelf and need to make sure the shelves are perfectly perpendicular to the sides. How can you use just a tape measure to guarantee a perfect 90° angle? 📐
You already know how the Pythagorean theorem helps you find a missing side in a right tria...
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Key Concepts & Vocabulary
TermDefinitionExample
Pythagorean TheoremIn 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 (legs).In a right triangle with legs of 3 cm and 4 cm, the hypotenuse is 5 cm because 3² + 4² = 9 + 16 = 25, and 5² = 25.
Converse of the Pythagorean TheoremIf the square of the length of the longest side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right triangle.A triangle with side lengths 5, 12, and 13 is a right triangle because 5² + 12² = 25 + 144 = 169, which is equal to 13².
HypotenuseThe longest side of a right triangle, located opposite the right angle. When applying the converse, 'c' mu...
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Core Formulas
Converse of the Pythagorean Theorem
Given a triangle with sides a, b, and c, where c is the longest side: If a² + b² = c², then the triangle is a right triangle.
Use this rule to prove that a triangle contains a 90-degree angle, making it a right triangle. This is a test for 'rightness'.
Pythagorean Inequality for Acute Triangles
Given a triangle with sides a, b, and c, where c is the longest side: If a² + b² > c², then the triangle is an acute triangle.
Use this rule when the sum of the squares of the two shorter sides is greater than the square of the longest side. This indicates all angles are less than 90 degrees.
Pythagorean Inequality for Obtuse Triangles
Given a triangle with sides a, b, and c, where c is the longest side: If a² + b² < c², then t...
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Challenging
A triangle has side lengths of x, x+7, and x+8. For which positive value of x is the triangle a right triangle?
A.3
B.4
C.5
D.7
Challenging
A rectangular prism has edge lengths of 3 cm, 4 cm, and 5 cm. Consider the triangle formed by an edge of length 5 cm, the diagonal of the 3x4 face, and the main space diagonal of the prism. Is this triangle a right triangle?
A.Yes, because 3² + 4² = 5².
B.Yes, because the sides form a right triangle when tested with the converse.
C.No, because the space diagonal is not perpendicular to the face diagonal.
D.It cannot be determined without knowing the angles.
Challenging
A student concludes a triangle with sides 7, 9, 11 is acute because 7²+9² = 130 and 11² = 121, and 130 > 121. What theorem or principle is the justification for this conclusion?
A.The Pythagorean Theorem
B.The Triangle Inequality Theorem
C.The Pythagorean Inequality for Acute Triangles
D.The Law of Cosines
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Frequently asked questions
What grade level is "Converse of the Pythagorean theorem"?
Converse of the Pythagorean theorem is a Grade 10 Mathematics lesson on ExcelOS.
What will I learn in Converse of the Pythagorean theorem?
You'll be able to: Identify the relationship between different customary units of length (inches, feet, yards, miles) with 100% accuracy; Solve word problems involving the conversion of customary units of length (inches, feet, yards, miles) with….
Is "Converse of the Pythagorean theorem" free to practice?
Yes. You can read the tutorial preview for free, and signing up for a free ExcelOS account unlocks the full tutorial and all practice questions with instant feedback.
How many practice questions are included with Converse of the Pythagorean theorem?
This lesson includes 25 practice questions across multiple difficulty levels, each with instant feedback and explanations.