Mathematics
Grade 8
15 min
Converse of pythagorean theorem: is it a right angled triangle?
Converse of pythagorean theorem: is it a right angled triangle?
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Introduction & Learning Objectives
Learning Objectives
Recall the Pythagorean Theorem and its components (legs and hypotenuse).
State the Converse of the Pythagorean Theorem.
Identify the longest side of a triangle as the potential hypotenuse.
Apply the Converse of the Pythagorean Theorem to determine if a given triangle is a right-angled triangle.
Perform calculations involving squaring numbers and comparing sums to verify the theorem.
Distinguish between triangles that are right-angled and those that are not using the theorem's converse.
Imagine you're building a bookshelf 📚. How can you be absolutely sure the corners are perfectly square (90 degrees) without a protractor?
In this lesson, we'll explore the Converse of the Pythagorean Theorem, a powerful tool that allows us to check if any t...
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Key Concepts & Vocabulary
TermDefinitionExample
Pythagorean TheoremA fundamental relationship in Euclidean geometry among the three sides of a right-angled 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).In a right triangle with legs of length 3 and 4, the hypotenuse is 5 because $3^2 + 4^2 = 9 + 16 = 25 = 5^2$.
Converse of the Pythagorean TheoremThe reverse statement of the Pythagorean Theorem. It states that if the square of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right-angled triangle.If a triangle has sides 6, 8, and 10, we check if $6^2 + 8^2 = 10^2$. Since $36 + 64 = 100$, and $10^2 = 100$, it is a right-angled triangle.
Right-Angl...
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Core Formulas
Pythagorean Theorem
$a^2 + b^2 = c^2$
Used to find an unknown side length in a *known* right-angled triangle, where 'a' and 'b' are the lengths of the legs, and 'c' is the length of the hypotenuse.
Converse of the Pythagorean Theorem
If $a^2 + b^2 = c^2$, then the triangle is a right-angled triangle.
Used to *determine* if a triangle with given side lengths 'a', 'b', and 'c' (where 'c' is the longest side) is a right-angled triangle. If the equation holds true, it's a right triangle; otherwise, it's not.
Identifying Non-Right Triangles
If $a^2 + b^2
eq c^2$, then the triangle is not a right-angled triangle.
This rule is the direct consequence of the converse. If the sum of the squares of...
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Challenging
Three square-shaped patios are connected at their corners to form a triangle. The areas of the patios are 25 m², 144 m², and 169 m². Is the triangle formed between them a right-angled triangle?
A.No, because 25 + 144 ≠169.
B.Yes, because 25 + 144 = 169.
C.No, because you need side lengths, not areas.
D.Yes, because the areas are perfect squares.
Challenging
A triangle is drawn on a coordinate plane with vertices at A(1, 2), B(4, 6), and C(8, 3). To determine if it is a right-angled triangle, you first need the lengths of the sides. The squared lengths are AB²=25, BC²=25, and AC²=50. Is triangle ABC a right-angled triangle?
A.No, because two of the sides are equal.
B.Yes, because 25 + 50 ≠25.
C.Yes, because 25 + 25 = 50.
D.No, because 25 + 25 ≠50.
Challenging
A triangle has side lengths of 10, 15, and 17. Using the converse of the Pythagorean theorem, what can be determined about this triangle?
A.It is a right-angled triangle because 10² + 15² is close to 17².
B.It is a right-angled triangle because 10² + 17² ≠15².
C.It is not a right-angled triangle because the sides are not a Pythagorean triple.
D.It is not a right-angled triangle because 10² + 15² ≠17².
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