Question

consider the triangle.

which statement is true about the lengths of the sides?

each side has a different length.
two sides have the same length, which is less than the length of the third side.
the three sides have the same length.
the sum of the lengths of two sides is equal to the length of the third side.

Explanation:

Identify the triangle type

The given triangle has angle measures of \(45^\circ\), \(45^\circ\), and a right angle (\(90^\circ\)). This is a \(45^\circ\)-\(45^\circ\)-\(90^\circ\) special right triangle, which is an isosceles right triangle.

Determine side length relationships

In an isosceles triangle, the sides opposite equal angles are equal in length.

• The two legs opposite the \(45^\circ\) angles are equal in length: \(a = b\).
• The hypotenuse \(c\) opposite the \(90^\circ\) angle is related to the legs by \(c = a\sqrt{2}\).
• Since \(\sqrt{2} \approx 1.414 > 1\), the hypotenuse is longer than each of the legs (\(c > a\)).
• Therefore, two sides have the same length, and this length is less than the length of the third side.

Evaluate the options

• "Each side has a different length" is false because two sides are equal.
• "Two sides have the same length, which is less than the length of the third side" is true because the two legs are equal, and each leg is shorter than the hypotenuse.
• "The three sides have the same length" is false because the hypotenuse is longer than the legs.
• "The sum of the lengths of two sides is equal to the length of the third side" is false by the triangle inequality theorem (the sum of any two side lengths must be strictly greater than the third).

Answer:

• Each side has a different length.
• Two sides have the same length, which is less than the length of the third side. (Correct answer)
• The three sides have the same length.
• The sum of the lengths of two sides is equal to the length of the third side.