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:
[tex]\[10^2 \ ? \ 11^2 + 15^2\][/tex]
[tex]\[100 \ ? \ 121 + 225\][/tex]
[tex]\[100 \ \textless \ 346\][/tex]

Conclusion:
This triangle is an acute triangle.

Which statement best summarizes Ella's work?

A. Ella's procedure and conclusion are correct.
B. Ella's procedure is correct, but her conclusion is incorrect.
C. Ella's procedure is incorrect, but her conclusion is correct.
D. Ella's procedure and conclusion are incorrect.



Answer :

To determine the type of triangle given side lengths 10, 11, and 15, let's go through the mathematical process step by step.

Firstly, label the sides of the triangle:
- [tex]\( a = 10 \)[/tex]
- [tex]\( b = 11 \)[/tex]
- [tex]\( c = 15 \)[/tex]
Typically, in a triangle's analysis, the longest side (15, in this case) is considered the hypotenuse if the triangle is a right triangle.

Step 1: Verify whether it's an acute, right, or obtuse triangle using the Pythagorean theorem:
- For a right triangle: [tex]\( a^2 + b^2 = c^2 \)[/tex]
- For an acute triangle: [tex]\( a^2 + b^2 > c^2 \)[/tex]
- For an obtuse triangle: [tex]\( a^2 + b^2 < c^2 \)[/tex]

Step 2: Calculate the squares of the side lengths:
- [tex]\( 10^2 = 100 \)[/tex]
- [tex]\( 11^2 = 121 \)[/tex]
- [tex]\( 15^2 = 225 \)[/tex]

Step 3: Compare sums of squares:
- Calculate [tex]\( a^2 + b^2 = 100 + 121 = 221 \)[/tex]
- Compare it with [tex]\( c^2 \)[/tex]:
- [tex]\( 221 \)[/tex] is less than [tex]\( 225 \)[/tex]

Since [tex]\( a^2 + b^2 < c^2 \)[/tex] (221 < 225), the triangle is an obtuse triangle.

Step 4: Review Ella's procedure and conclusion:
- Ella should have compared [tex]\( 100 + 121 \)[/tex] with [tex]\( 225 \)[/tex], but she incorrectly added 11 to 15, leading to a false result.
- She concluded that [tex]\( 346 > 100 \)[/tex], which misled her to declare the triangle acute.

Therefore, the best summary of Ella's work is:

Ella's procedure is correct, but her conclusion is incorrect.