which of these four sets of side lengths will form a right triangle?

set 1: 6 cm, 7 cm, $\sqrt{12}$ cm
set 2: 8 in., $\sqrt{29}$ in., $\sqrt{35}$ in.
set 3: $\sqrt{3}$ mm, 4 mm, $\sqrt{5}$ mm
set 4: 9 ft, $\sqrt{26}$ ft, 6 ft

- set 1
- set 2
- set 3
- set 4

Question

which of these four sets of side lengths will form a right triangle?

set 1: 6 cm, 7 cm, $\sqrt{12}$ cm
set 2: 8 in., $\sqrt{29}$ in., $\sqrt{35}$ in.
set 3: $\sqrt{3}$ mm, 4 mm, $\sqrt{5}$ mm
set 4: 9 ft, $\sqrt{26}$ ft, 6 ft

- set 1
- set 2
- set 3
- set 4

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Pythagorean Theorem" ], "new_concepts": [], "current_concepts": [ "Pythagorean Theorem", "Converse of the Pythagorean Theorem" ] } ## Identify the longest side (hypotenuse candidate) for each set $$ \begin{aligned} &\text{Set 1: } 6, 7, \sqrt{12} \approx 3.46 \implies c = 7 \ &\text{Set 2: } 8, \sqrt{29} \approx 5.39, \sqrt{35} \approx 5.92 \implies c = 8 \ &\text{Set 3: } \sqrt{3} \approx 1.73, 4, \sqrt{5} \approx 2.24 \implies c = 4 \ &\text{Set 4: } 9, \sqrt{26} \approx 5.10, 6 \implies c = 9 \end{aligned} $$ ## Test the Pythagorean equation $a^2 + b^2 = c^2$ for each set $$ \begin{aligned} &\text{Set 1: } 6^2 + (\sqrt{12})^2 = 36 + 12 = 48 eq 7^2 \ (49) \ &\text{Set 2: } (\sqrt{29})^2 + (\sqrt{35})^2 = 29 + 35 = 64 = 8^2 \ (64) \ &\text{Set 3: } (\sqrt{3})^2 + (\sqrt{5})^2 = 3 + 5 = 8 eq 4^2 \ (16) \ &\text{Set 4: } 6^2 + (\sqrt{26})^2 = 36 + 26 = 62 eq 9^2 \ (81) \end{aligned} $$ ## Determine the correct set $$ \text{Set 2 forms a right triangle because } (\sqrt{29})^2 + (\sqrt{35})^2 = 8^2. $$

Set 1

Set 2

Set 3

Set 4

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

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QUESTION IMAGE

Question

Explanation:

Identify the longest side (hypotenuse candidate) for each set

Test the Pythagorean equation \(a^2 + b^2 = c^2\) for each set

Determine the correct set

Answer: