7.4. In isosceles triangle $ABC$, the base $AC$ is equal to $x$, and the lateral side is 12. On the ray $AC$, point $D$ is marked such that $AD=24$. A perpendicular $DE$ is dropped from point $D$ to line $AB$. Find $x$ if it is known that $BE=6$.

![](https://cdn.mathpix.com/cropped/2024_05_06_0b1749f4383e44f39544g-7.jpg?height=357&width=560&top_left_y=490&top_left_x=725)

Answer, option 1. 13.

Answer, option 2. 17.

Answer, option 3. 14.

Answer, option 4. 18.
Solution Option 1. In triangle $A E D$, the angle at vertex $E$ is a right angle. In a right triangle, it is often useful to draw the median from the vertex of the right angle. If $E M$ is the median, then it is equal to half the hypotenuse $A D$, that is, $E M=M A=M D=10$.

![](https://cdn.mathpix.com/cropped/2024_05_06_0b1749f4383e44f39544g-7.jpg?height=460&width=723&top_left_y=1575&top_left_x=678)

Consider the isosceles triangles $A M E$ and $A B C$. Their lateral sides are equal to 10, and the angles at the base in each of them are equal to the angle $B A C$. Therefore, the angles at the vertices in these triangles are also equal, and thus the triangles themselves are equal by the first criterion. Consequently, $A C=A E=A B+B E=13$.