Understanding the Isosceles Triangle
To find the base of an isosceles triangle given the area and one side, we can apply the formula for the area of a triangle and some basic geometry.
Given Data
- Area (A) = 12 cm²
- Length of one side (s) = 5 cm
Area Formula
The area of a triangle can be calculated using the formula:
Area = (base * height) / 2
In this case, we need to express the height in terms of the base.
Height Calculation
For an isosceles triangle:
- Let the base be "b"
- The height (h) can be found using the Pythagorean theorem.
The height splits the triangle into two right triangles, each with:
- Half of the base = b/2
- Hypotenuse = s = 5 cm
Using the Pythagorean theorem:
h² + (b/2)² = s²
h² + (b/2)² = 5²
Area Substitution
From the area formula:
12 = (b * h) / 2
This can be rearranged to:
h = (24 / b)
Now substituting h into the Pythagorean theorem:
(24 / b)² + (b/2)² = 25
Solving for Base (b)
1. Expand and simplify the equation:
- (576 / b²) + (b² / 4) = 25
2. Multiply through by b² to eliminate the fraction:
- 576 + (b⁴ / 4) = 25b²
3. Rearranging gives a quadratic in terms of b²:
- b⁴ - 100b² + 2304 = 0
4. Using the quadratic formula, the solutions can be found to determine the base (b).
Conclusion
By solving the quadratic equation, you can find the value of the base of the triangle. The calculated base will ensure the area meets the given requirement of 12 cm², confirming the dimensions of your isosceles triangle.