Question

ella’s geometry teacher asked each student to devise a problem and write out its solution. here is ella’s work: a triangle has side lengths of 10, 11, and 15. what type of triangle is it? procedure: 10² ?? 11² + 15² 100 ?? 121 + 225 100 < 346 conclusion: this triangle is an acute triangle. which statement best summarizes ellas work? ○ ella’s procedure and conclusion are correct. ○ ella’s procedure is correct, but her conclusion is incorrect. ○ ella’s procedure is incorrect, but her conclusion is correct. ○ ella’s procedure and conclusion are incorrect.

Explanation:

Step1: Recall triangle type rules

To determine triangle type (acute, right, obtuse), use the Pythagorean inequality. For sides \(a \leq b \leq c\):

• Right: \(a^2 + b^2 = c^2\)
• Acute: \(a^2 + b^2 > c^2\)
• Obtuse: \(a^2 + b^2 < c^2\)

Step2: Identify longest side

Given sides 10, 11, 15. Longest side \(c = 15\), so \(a = 10\), \(b = 11\).

Step3: Apply inequality

Calculate \(a^2 + b^2 = 10^2 + 11^2 = 100 + 121 = 221\), \(c^2 = 15^2 = 225\).
Compare: \(221 < 225\) (so \(a^2 + b^2 < c^2\)), meaning the triangle is obtuse.

Step4: Analyze Ella’s work

• Procedure: Ella used \(10^2\) vs \(11^2 + 15^2\) (wrong: should compare \(a^2 + b^2\) vs \(c^2\), not \(a^2\) vs \(b^2 + c^2\)).
• Conclusion: She said acute, but it’s obtuse (incorrect).

Answer:

Ella’s procedure and conclusion are incorrect.