Question

shelf is designed so it will fit in a 90º corner between two walls. The shelf has dimensions, rounded to the nearest tenth, as shown.

A triangle has side lengths 11 centimeters, 11 centimeters, and hypotenuse of StartRoot 242 EndRoot centimeters.

[Not drawn to scale]


Will the shelf fit snugly in a 90º corner?
Yes, 2 (11) squared = StartRoot 242 EndRoot squared.
No, (11 + 11) squared not-equals StartRoot 242 EndRoot squared.
No, 11 squared + StartRoot 242 EndRoot squared not-equals 11 squared.
No, 11 squared not-equals StartRoot 242 EndRoot squared.

Answer 1

To determine if the shelf will fit snugly in a 90º corner, we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (\(c\)) is equal to the sum of the squares of the lengths of the other two sides (\(a\) and \(b\)).

Given:
- Two sides (legs) of the triangle: \(a = 11\) cm and \(b = 11\) cm.
- Hypotenuse (\(c\)): \(\sqrt{242}\) cm.

According to the Pythagorean theorem:
\[ c^2 = a^2 + b^2 \]

Calculating:
1. Calculate \(a^2\) and \(b^2\):
\[
a^2 = 11^2 = 121
\]
\[
b^2 = 11^2 = 121
\]

2. Find the sum of \(a^2\) and \(b^2\):
\[
a^2 + b^2 = 121 + 121 = 242
\]

3. Calculate \(c^2\):
\[
c = \sqrt{242} \implies c^2 = 242
\]

Now we compare \(c^2\) to \(a^2 + b^2\):
\[
c^2 = 242
\]
\[
a^2 + b^2 = 242
\]

Since both sides are equal:
\[
c^2 = a^2 + b^2
\]
This verifies that the relationship holds true.

Therefore, the shelf will fit snugly in a 90º corner, and the correct answer is:
**Yes, \( 2(11)^2 = \sqrt{242}^2.**