A triangle has two sides measuring 20 feet and one side measuring 9 feet. What kind of triangle is it?

1. Break Down the Problem

To classify the triangle based on the lengths of its sides (20 feet, 20 feet, and 9 feet), we will analyze the properties of the triangle formed by these sides.

2. Relevant Concepts

A triangle can be classified as:

• Equilateral: All three sides are equal.
• Isosceles: Exactly two sides are equal.
• Scalene: All sides are different.

We can also determine if it is an acute, right, or obtuse triangle using the Pythagorean theorem for comparison when the triangle is scalene.

3. Analysis and Detail

1. Check Side Lengths:

   • The sides are: 20 feet, 20 feet, and 9 feet.
   • Two sides (20 feet) are equal, indicating that it is at least an isosceles triangle.

2. Classifying the Triangle:

   • Since two sides are equal, the triangle is classified as Isosceles.

3. Determine Angle Classification:

   • Using the triangle inequality theorem, we check if $a + b > c$:
     • $20 + 20 > 9$, which is true.
   • Comparing the squares of the sides to determine whether it is acute or obtuse:
     • $a = 20$, $b = 20$, $c = 9$
     • $a^2 + b^2 = 20^2 + 20^2 = 400 + 400 = 800$
     • $c^2 = 9^2 = 81$
     • Since $a^2 + b^2 > c^2$, the triangle is acute as well.

4. Verify and Summarize

The triangle is confirmed to be isosceles because two side lengths are equal (20 feet). Further, it is acute since the sum of the squares of the two equal sides is greater than the square of the third side.

Final Answer

The triangle is classified as an isosceles acute triangle.