What is the area of the given figure?

### Concept & Composite Area To find the area of a composite figure, break it down into simpler geometric shapes (rectangles and triangles). The figure consists of a central rectangle and two right-angled triangles attached at the ends. We use the Pythagorean theorem to find missing side lengths. Pythagorean theorem: $$a^2 + b^2 = c^2$$ ### Step-by-Step Solution * **Given:** A central rectangle with length 12 cm. Two right-angled triangles at the ends with legs of 4 cm and 6 cm. * **Analyze the Triangles:** The right angle symbol indicates the legs are 4 cm and 6 cm. Area of one triangle = $\frac{1}{2} \times 4 \times 6 = 12 \text{ cm}^2$. Total area for two triangles = $12 \times 2 = 24 \text{ cm}^2$. * **Find Rectangle Width:** The hypotenuse of these right triangles forms the width of the central rectangle. $\text{Width} = \sqrt{4^2 + 6^2} = \sqrt{16 + 36} = \sqrt{52}$ cm. $\sqrt{52} \approx 7.21$ cm (or typically approximated as 7.2 in such problems). * **Calculate Rectangle Area:** Area = $\text{Length} \times \text{Width} = 12 \times \sqrt{52} \approx 12 \times 7.2 = 86.4 \text{ cm}^2$. * **Total Area:** Total Area = Rectangle Area + Triangles Area = $86.4 + 24 = 110.4 \text{ cm}^2$. ### Exam Strategy & Shortcut Recognize that $\sqrt{52}$ is slightly larger than $\sqrt{49}$ (which is 7). $12 \times 7 = 84$. Total area is slightly more than $84 + 24 = 108$. Looking at the options, $110.4 \text{ cm}^2$ is the only logical choice that perfectly aligns with a standard approximation of $\sqrt{52} \approx 7.2$. ### Common Pitfall Misidentifying the hypotenuse. If one assumes the 6 cm side is the hypotenuse instead of a leg, the calculations will lead to an incorrect and missing option. Always pay close attention to where the right-angle square symbol is placed. ### Final Answer Therefore, the correct answer is **110.4 cm²**.