To solve the problem, we need to find the value of $x$ using the properties of triangles and trigonometry.
1. Observing the Triangle Configuration:
In the given figure, triangles $\triangle ABC$ and $\triangle DEC$ share angle $\angle C$, and angles $\angle A = \angle E$ (both marked as equal), which means the triangles are similar by AA similarity.
2. Using Property of Similar Triangles:
From similar triangles $\triangle ABC \sim \triangle DEC$, the corresponding sides are in proportion:
$\frac{AB}{DE} = \frac{AC}{DC}$
3. Substituting Given Values:
$AB = 24$
$DE = 12$
$BC = 22$
Let $x = DC$
Using similar triangles, we write:
$\frac{24}{12} = \frac{22}{x}$
4. Solving the Proportion:
$\frac{24}{12} = 2$, so:
$2 = \frac{22}{x}$
$x = \frac{22}{2} = 11$
Final Answer:
The value of $x$ is $ \mathbf{11} $.
In the given figure, the numbers associated with the rectangle, triangle, and ellipse are 1, 2, and 3, respectively. Which one among the given options is the most appropriate combination of \( P \), \( Q \), and \( R \)?
Find the number of triangles in the given figure.
A regular dodecagon (12-sided regular polygon) is inscribed in a circle of radius \( r \) cm as shown in the figure. The side of the dodecagon is \( d \) cm. All the triangles (numbered 1 to 12 in the figure) are used to form squares of side \( r \) cm, and each numbered triangle is used only once to form a square. The number of squares that can be formed and the number of triangles required to form each square, respectively, are: