Question

what is the value of x?
○ $\frac{1}{3}\sqrt{2}$ units
○ $\frac{1}{2}\sqrt{3}$ units
○ $2\sqrt{3}$ units
○ $3\sqrt{2}$ units

Explanation:

Step1: Identify the triangle type

The left side of the large triangle is split into two segments of length 3, so the total length is \(3 + 3 = 6\)? Wait, no, looking at the diagram, the two segments are both 3, and there are right angles. Wait, actually, this is a geometric mean (altitude-on-hypotenuse) theorem? Wait, no, maybe it's an isosceles triangle with a perpendicular from the apex. Wait, no, the right angles: the two smaller triangles are right triangles, and the large triangle? Wait, maybe it's a triangle where the altitude to the hypotenuse creates two similar triangles, but here, the left side is 3 + 3 = 6? Wait, no, the left side has two segments of 3, so the total length from top to bottom is 3 + 3 = 6? Wait, no, the diagram shows a triangle with a vertical side split into two 3s, and two right angles. Wait, maybe it's a triangle where the hypotenuse of the top right triangle is \(x\), and the vertical side is 3, and the other side? Wait, no, let's re-examine.

Wait, the diagram: there's a triangle with a vertical segment of length 3 (top) and 3 (bottom), so total vertical length 6? No, wait, the left side is two 3s, so the total length from the top vertex to the bottom vertex is 3 + 3 = 6? And there are two right angles, so the line from the right vertex to the left side is an altitude? Wait, no, maybe it's a triangle where the two legs of the top right triangle are 3 and \(x\), and the bottom triangle? Wait, no, maybe it's a 45-45-90 triangle? Wait, no, the options have \(3\sqrt{2}\). Wait, if the vertical side is 3 + 3 = 6? No, wait, the left side is split into two 3s, so the length from the top to the middle is 3, middle to bottom is 3. The right side has a right angle at the middle, so the top triangle is a right triangle with legs 3 and \(x\), and the bottom triangle? Wait, no, maybe the large triangle is isosceles with legs 6? No, that doesn't make sense. Wait, maybe it's a triangle where the altitude is 3, and the hypotenuse of the top triangle is \(x\), and the base is 3? No, wait, the key is the geometric mean or maybe a right triangle with legs 3 and 3, so hypotenuse \(3\sqrt{2}\). Wait, if the top triangle is a right triangle with legs 3 and 3, then the hypotenuse \(x\) would be \(\sqrt{3^2 + 3^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2}\). Yes, that makes sense. So the top right triangle is a right triangle with legs 3 and 3, so hypotenuse \(x = \sqrt{3^2 + 3^2} = 3\sqrt{2}\).

Step2: Apply Pythagorean theorem

For a right triangle with legs \(a = 3\) and \(b = 3\), the hypotenuse \(x\) is given by \(x = \sqrt{a^2 + b^2}\).

Substitute \(a = 3\) and \(b = 3\):

\(x = \sqrt{3^2 + 3^2} = \sqrt{9 + 9} = \sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}\).

Answer:

\(3\sqrt{2}\) units (the last option: \(3\sqrt{2}\) units)