1. **State the problem:** We have a triangle with base length $x - 8$ cm and height $2x$ cm. The area is given as 20 cm$^2$. We need to find the values of the base length and height. 2. **Recall the formula for the area of a triangle:** $$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$$ 3. **Substitute the given expressions:** $$20 = \frac{1}{2} \times (x - 8) \times 2x$$ 4. **Simplify the equation:** $$20 = (x - 8) \times x$$ $$20 = x^2 - 8x$$ 5. **Rewrite as a quadratic equation:** $$x^2 - 8x - 20 = 0$$ 6. **Solve the quadratic equation using the quadratic formula:** $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=1$, $b=-8$, and $c=-20$. Calculate the discriminant: $$\Delta = (-8)^2 - 4 \times 1 \times (-20) = 64 + 80 = 144$$ Calculate the roots: $$x = \frac{8 \pm \sqrt{144}}{2} = \frac{8 \pm 12}{2}$$ 7. **Find the two possible values for $x$:** - $$x = \frac{8 + 12}{2} = \frac{20}{2} = 10$$ - $$x = \frac{8 - 12}{2} = \frac{-4}{2} = -2$$ 8. **Check for valid solution:** Since the base length is $x - 8$, it must be positive. - For $x=10$, base = $10 - 8 = 2$ cm (valid) - For $x=-2$, base = $-2 - 8 = -10$ cm (not valid) 9. **Calculate the height for $x=10$:** $$\text{height} = 2x = 2 \times 10 = 20 \text{ cm}$$ **Final answer:** - Base length = 2 cm - Height = 20 cm