https://file.aiquickdraw.com/mathbot/file/1727062536404qsekc4hy.jpg

To solve this problem, let's break down the given information:

We are dealing with a triangle that has labeled sides and angles:
- One right triangle has angles of \(30^\circ\), \(60^\circ\), and \(90^\circ\).
- There is another triangle involved with a \(45^\circ\), \(45^\circ\), and \(90^\circ\) angle.

We need to find the exact values of the variables \(h\), \(b\), \(c\), and \(f\). For now, the problem is asking for \(h\), the base of the 30-60-90 triangle.

### Step 1: Use the properties of a 30-60-90 triangle
In a 30-60-90 triangle, the side ratios are:
- Short leg (opposite \(30^\circ\)) = \(x\)
- Long leg (opposite \(60^\circ\)) = \(x\sqrt{3}\)
- Hypotenuse = \(2x\)

Let the short leg be \(h\). From the image, \(b\) is the longer leg, and \(c\) is the hypotenuse.

### Step 2: Solve for \(h\)
Assume that the hypotenuse \(c\) of this triangle is given as part of the triangle involving \(f\) and \(c\) from the \(45^\circ\)-\(45^\circ\)-\(90^\circ\) triangle. We can relate these triangles via the fact that in a 30-60-90 triangle, the short leg \(h\) can be expressed in terms of the hypotenuse \(c\) using the ratio \(h = \frac{c}{2}\).

Once we have the hypotenuse \(c\) value (likely determined from the other triangle's geometry), we can directly find \(h\).

Would you like to proceed with finding the value of \(h\), or would you like more explanation on the other variables like \(b\) or \(f\)? Feel free to ask for any specific steps.

Find the exact value of each part labeled with a variable in the following figure. (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)

Learn how to solve for the variables h, b, c, and f in a diagram containing a 30-60-90 triangle and a 45-45-90 triangle. This problem involves using key properties and ratios of special right triangles. Ideal for students in Grades 9-11.

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