question 24 of 25

a triangle with sides of lengths 10, 24, and 27 is a right triangle.

a. true

b. false

Question

question 24 of 25

a triangle with sides of lengths 10, 24, and 27 is a right triangle.

a. true

b. false

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Pythagorean Theorem" ], "new_concepts": [ "Pythagorean Theorem Converse" ], "current_concepts": [ "Pythagorean Theorem", "Pythagorean Theorem Converse" ] } ## Identify the side lengths We are given a triangle with side lengths $a = 10$, $b = 24$, and $c = 27$. The longest side, which would be the hypotenuse if it is a right triangle, is $c = 27$. ## Apply the Pythagorean Theorem Using the **Pythagorean Theorem** knowledge point $$ \begin{aligned} a^2 + b^2 &= 10^2 + 24^2 \ &= 100 + 576 \ &= 676 \end{aligned} $$ ## Calculate the square of the longest side We calculate the square of the longest side $c$: $$ \begin{aligned} c^2 &= 27^2 \ &= 729 \end{aligned} $$ ## Compare the values We compare the sum of the squares of the two shorter sides to the square of the longest side: $$ \begin{aligned} a^2 + b^2 &= 676 \ c^2 &= 729 \ 676 & eq 729 \end{aligned} $$ Since $a^2 + b^2 eq c^2$, the triangle does not satisfy the converse of the Pythagorean theorem and is not a right triangle. Therefore, the statement is false.

A. True

B. False

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "Pythagorean Theorem Converse" ] }

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QUESTION IMAGE

Question

Explanation:

Identify the side lengths

Apply the Pythagorean Theorem

Calculate the square of the longest side

Compare the values

Answer: