The perimeter of an isosceles triangle is equal to 14 cm and the lateral side is to the base in the ratio $5 : 4$. The area of the triangle is

### Concept & Isosceles Triangle Properties An isosceles triangle features two equal lateral sides. By drawing a perpendicular altitude straight down to the base, you split the triangle into two identical right-angled triangles, allowing you to use the Pythagorean theorem to easily find the height. Area of a triangle: $$A = \frac{1}{2} \times \text{base} \times \text{height}$$ ### Step-by-Step Solution * **Given:** The total perimeter is 14 cm. The ratio of one lateral side to the base is $5 : 4$. Because it is isosceles, the two lateral sides are equal. Let the sides be $5x$, $5x$, and the base be $4x$. * **Find Side Lengths:** $\text{Perimeter} = 5x + 5x + 4x = 14x$ $14x = 14 \Rightarrow x = 1$ The actual side lengths are exactly 5 cm, 5 cm, and 4 cm. * **Find Height:** Drop a perpendicular line from the top vertex down to the base. This perfectly bisects the 4 cm base into two 2 cm segments. We now have a right triangle with a hypotenuse of 5 and a base of 2. $h^2 + 2^2 = 5^2$ $h^2 + 4 = 25$ $h^2 = 21 \Rightarrow h = \sqrt{21}\text{ cm}$. * **Calculate Area:** $A = \frac{1}{2} \times \text{total base} \times \text{height}$ $A = \frac{1}{2} \times 4 \times \sqrt{21} = 2\sqrt{21}\text{ cm}^2$. ### Exam Strategy & Shortcut Quickly resolve the ratio $5:5:4$ to sum to 14, meaning the sides are identically 5, 5, and 4. Knowing the altitude bisects the base to 2, mentally calculate $h = \sqrt{25 - 4} = \sqrt{21}$. Area is simply $\text{half-base} \times \text{height} = 2 \times \sqrt{21}$. ### Common Pitfall Misinterpreting the phrasing "lateral side is to the base" as meaning there is only one lateral side involved, leading to a flawed perimeter equation like $5x + 4x = 14$. Always remember a triangle must have three sides. ### Final Answer Therefore, the correct answer is **$2\sqrt{21}\text{ cm}^2$**.