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.
