Final answer:
The perimeter of an isosceles triangle with an area of 56 square units and a base that is one third of the height is approximately 28.4 units, none of the options provided in the question matches this result.
Step-by-step explanation:
To find the perimeter of an isosceles triangle with an area of 56 square units where the base is one third of the height, we can start by using the formula for the area of a triangle: Area = 1/2 × base × height.
Since the area is given as 56 square units, and we know that the base (b) is one third of the height (h), we can express the base as b = h/3. Substituting into the area formula gives us:
Solving for h, we find that h = 12 units and b = 4 units. To find the length of the equal sides, we can use the Pythagorean theorem because the equal sides and the height form two right triangles. If the equal sides are represented by s, then:
- s^2 = h^2 + (b/2)^2
- s^2 = 12^2 + 2^2
- s^2 = 144 + 4
- s = √148 ≈ 12.2 units
The perimeter (P) is the sum of the base and twice the length of the equal sides:
- P = b + 2s
- P = 4 + 2 × 12.2
- P = 4 + 24.4
- P ≈ 28.4 units
Therefore, the perimeter is approximately 28.4 units, which is not one of the given options, suggesting a possible error in the question or the options provided. We rounded all decimals to one decimal place throughout the calculations.