To determine the type of triangle given side lengths of 10, 11, and 15, Ella correctly follows the procedure to verify if a triangle is acute. Here is a step-by-step explanation:
1. Procedure Overview:
- We need to check the relationship between the squares of the side lengths of the triangle. Specifically, for a triangle with sides [tex]\( a, b, \)[/tex] and [tex]\( c \)[/tex], the triangle is acute if for each permutation of the sides, the square of the longest side is less than the sum of the squares of the other two sides.
2. Assigning Side Lengths:
- Let [tex]\( a = 10 \)[/tex]
- Let [tex]\( b = 11 \)[/tex]
- Let [tex]\( c = 15 \)[/tex]
3. Compare Squares:
- First, we calculate the square of each side.
[tex]\[
10^2 = 100
\][/tex]
[tex]\[
11^2 = 121
\][/tex]
[tex]\[
15^2 = 225
\][/tex]
- Next, we sum the squares of the two smaller sides and compare this to the square of the longest side:
[tex]\[
11^2 + 15^2 = 121 + 225 = 346
\][/tex]
4. Inequality Verification:
- According to the rule for an acute triangle:
[tex]\[
a^2 < b^2 + c^2
\][/tex]
- Substitute the values and check:
[tex]\[
100 < 346
\][/tex]
- This inequality holds true.
5. Conclusion:
- Since [tex]\( 100 < 346 \)[/tex], the triangle is indeed acute.
Therefore, the statement that best summarizes Ella's work is:
Ella's procedure and conclusion are correct.
