Based on the image, EF || TR, TR - EF = 14, ET = 13, FR = 15, what is the area of trapezoid TRFE?

Okay, let's solve this geometry problem step by step.

We are given a trapezoid TRFE with EF || TR. Also, we know that TR - EF = 14, ET = 13, and FR = 15. We need to find the area of the trapezoid.

Since a circle can be inscribed in the trapezoid TRFE, the sum of the lengths of the sides ET and FR must equal the sum of the lengths of the bases EF and TR. Therefore:

$$ET + FR = EF + TR$$

We are given that ET = 13 and FR = 15, so:

$$13 + 15 = EF + TR$$

$$28 = EF + TR$$

We also know that TR - EF = 14. We can use this information to solve for TR and EF. Let's solve this system of equations:

$$\begin{cases} TR + EF = 28 \\ TR - EF = 14 \end{cases}$$

Adding the two equations, we get:

$$2 * TR = 42$$

$$TR = 21$$

Now we can find EF:

$$EF = 28 - TR = 28 - 21 = 7$$

So, TR = 21 and EF = 7.

To find the area of the trapezoid, we need to find its height. Let's denote the height by h. We can drop perpendiculars from points E and F to the base TR, forming two right triangles. Let's denote the feet of these perpendiculars on TR as T' and R', respectively. Then, ET' = x and FR' = y, and x + y = TR - EF = 14. Also, we have two right triangles: ETT' and FRR'. In both triangles, the height is h. Using the Pythagorean theorem for triangles ETT' and FRR', we have:

$$h^2 + x^2 = 13^2 = 169$$

$$h^2 + y^2 = 15^2 = 225$$

Subtracting the first equation from the second equation, we get:

$$y^2 - x^2 = 225 - 169 = 56$$

$$(y - x)(y + x) = 56$$

We know that y + x = 14, so:

$$(y - x) * 14 = 56$$

$$y - x = 4$$

Now we have a system of equations:

$$\begin{cases} y + x = 14 \\ y - x = 4 \end{cases}$$

Adding the two equations, we get:

$$2y = 18 \Rightarrow y = 9$$

Then:

$$x = 14 - y = 14 - 9 = 5$$

Now we can find the height h using either of the Pythagorean equations. Let's use the first one:

$$h^2 + 5^2 = 169$$

$$h^2 + 25 = 169$$

$$h^2 = 144$$

$$h = 12$$

Now we can find the area of the trapezoid using the formula:

$$S = \frac{EF + TR}{2} * h = \frac{7 + 21}{2} * 12 = \frac{28}{2} * 12 = 14 * 12 = 168$$

So the area of the trapezoid TRFE is 168.

Ответ: 168