Question

1. x = ??? (show your work below!!)

(image of a right triangle with legs 3, 3, angles 45°, 45°, right angle, and hypotenuse x)

1. x = ??? (show your work below!!)

(image of a right triangle with legs 4, 4, angles 45°, 45°, right angle, and hypotenuse x)

1. x = ??? (show your work below!!)

(image of a right triangle with legs 2, 2, angles 45°, 45°, right angle, and hypotenuse x)

Explanation:

Problem 2:

Step1: Identify Triangle Type

The triangle is a right - isosceles triangle (two angles \(45^{\circ}\), one right angle). In a right - isosceles triangle, the legs are equal, and we can use the Pythagorean theorem \(c^{2}=a^{2}+b^{2}\), where \(a = b = 3\) and \(c=x\).

Step2: Apply Pythagorean Theorem

Substitute \(a = 3\) and \(b = 3\) into the formula: \(x^{2}=3^{2}+3^{2}=9 + 9=18\). Then \(x=\sqrt{18}=\sqrt{9\times2}=3\sqrt{2}\).

Step1: Identify Triangle Type

The triangle is a right - isosceles triangle (two angles \(45^{\circ}\), one right angle). The legs \(a = b = 4\), and we use the Pythagorean theorem \(c^{2}=a^{2}+b^{2}\) with \(c = x\).

Step2: Apply Pythagorean Theorem

Substitute \(a = 4\) and \(b = 4\) into the formula: \(x^{2}=4^{2}+4^{2}=16 + 16 = 32\). Then \(x=\sqrt{32}=\sqrt{16\times2}=4\sqrt{2}\).

Step1: Identify Triangle Type

The triangle is a right - isosceles triangle (two angles \(45^{\circ}\), one right angle). The legs \(a = b = 2\), and we use the Pythagorean theorem \(c^{2}=a^{2}+b^{2}\) with \(c=x\).

Step2: Apply Pythagorean Theorem

Substitute \(a = 2\) and \(b = 2\) into the formula: \(x^{2}=2^{2}+2^{2}=4 + 4=8\). Then \(x=\sqrt{8}=\sqrt{4\times2}=2\sqrt{2}\).

Answer:

\(3\sqrt{2}\)

Problem 3: