The area of the triangle is 35 square meters. Find the dimensions of the triangle, given the height is (g - 11) meters and the base is (g - 8) meters.

Steps to Solve

1. Write the formula for the area of a triangle

The area $A$ of a triangle is given by the formula: $$ A = \frac{1}{2} \cdot \text{base} \cdot \text{height} $$

2. Substitute the given values into the formula

We are given that the area $A = 35$, the base is $(g-8)$, and the height is $(g-11)$. Substituting these values into the formula, we get: $$ 35 = \frac{1}{2} (g-8)(g-11) $$

3. Simplify the equation and expand the terms

Multiply both sides of the equation by 2: $$ 70 = (g-8)(g-11) $$ Expand the right side of the equation: $$ 70 = g^2 - 11g - 8g + 88 $$ $$ 70 = g^2 - 19g + 88 $$

4. Rearrange the equation into a quadratic equation

Subtract 70 from both sides of the equation to set it equal to zero: $$ 0 = g^2 - 19g + 88 - 70 $$ $$ 0 = g^2 - 19g + 18 $$

5. Solve the quadratic equation

We can solve this quadratic equation by factoring. We are looking for two numbers that multiply to 18 and add up to -19. These numbers are -1 and -18: $$ 0 = (g - 1)(g - 18) $$ Setting each factor equal to zero gives us two possible solutions for $g$: $$ g - 1 = 0 \Rightarrow g = 1 $$ $$ g - 18 = 0 \Rightarrow g = 18 $$

6. Check the validity of the solutions

If $g = 1$, then the height would be $g - 11 = 1 - 11 = -10$ meters, and the base would be $g - 8 = 1 - 8 = -7$ meters. Since the dimensions of a triangle cannot be negative, $g = 1$ is not a valid solution. If $g = 18$, then the height would be $g - 11 = 18 - 11 = 7$ meters, and the base would be $g - 8 = 18 - 8 = 10$ meters. These are both positive, so $g = 18$ is a valid solution.

7. Find the dimensions of the triangle Using $g=18$, the height is $18-11 = 7$ meters and the base is $18-8 = 10$ meters.