Question

ellas geometry teacher asked each student to devise a problem and write out its solution. here is ellas work:
a triangle has side lengths of 10, 11, and 15. what type of triangle is it?
procedure:
$10^2 ?? 11^2 + 15^2$
$100 ?? 121 + 225$
$100 < 346$
conclusion:
this triangle is an acute triangle
which statement best summarizes ellas work?
○ ellas procedure and conclusion are correct.
○ ellas procedure is correct, but her conclusion is incorrect.
○ ellas procedure is incorrect, but her conclusion is correct.
○ ellas procedure and conclusion are incorrect.

Explanation:

Step1: Identify correct comparison rule

To classify a triangle by side lengths, we use the relationship with the square of the longest side. For sides $a \leq b \leq c$:

• If $c^2 < a^2 + b^2$: Acute triangle
• If $c^2 = a^2 + b^2$: Right triangle
• If $c^2 > a^2 + b^2$: Obtuse triangle

Here, the longest side is 15, so we compare $15^2$ to $10^2 + 11^2$.

Step2: Calculate correct side squares

$$15^2 = 225$$
$$10^2 + 11^2 = 100 + 121 = 221$$

Step3: Evaluate Ella's procedure

Ella incorrectly used the shortest side ($10^2$) for the comparison instead of the longest side. Her procedure is wrong.

Step4: Evaluate Ella's conclusion

Since $225 > 221$, the triangle is obtuse, so Ella's conclusion of "acute triangle" is also incorrect.

Answer:

Ella's procedure and conclusion are incorrect.