Determine the maximum area of a triangle in square centimetres, if the sum of its base and its height is 10 cm. 1.) The sum of the

## Step 1: Expressing height in terms of base<br />### Let 'b' be the base and 'h' be the height of the triangle. We are given that $b + h = 10$. We can express the height in terms of the base: $h = 10 - b$.<br />## Step 2: Area of the triangle<br />### The area of a triangle is given by the formula $A = \frac{1}{2}bh$. Substituting $h = 10 - b$, we get $A = \frac{1}{2}b(10 - b) = 5b - \frac{1}{2}b^2$.<br />## Step 3: Finding the maximum area<br />### To find the maximum area, we can complete the square or find the vertex of the parabola represented by the area function. The x-coordinate of the vertex of a parabola in the form $y = ax^2 + bx + c$ is given by $x = -\frac{b}{2a}$. In our case, $A = -\frac{1}{2}b^2 + 5b$, so $a = -\frac{1}{2}$ and $b = 5$. Thus, $b = -\frac{5}{2(-\frac{1}{2})} = 5$. Substituting $b=5$ into the area formula: $A = \frac{1}{2}(5)(10-5) = \frac{1}{2}(5)(5) = 12.5$.