Question: User: 60)\nx\n5\n10\n11\n\nA = ________\nP = ________\n\nTrapezoid near the top-center of the page, with top base labeled $x$, left slanted side labeled $5$, right slanted side labeled $10$, bottom base labeled $11$, and a dashed vertical altitude from the top-left vertex to the bottom base with a right-angle marker at the foot.

1. **State the problem:** We have a trapezoid with bases $x$ (top) and $11$ (bottom), and legs $5$ (left) and $10$ (right). We need to find the area $A$ and perimeter $P$. 2. **Recall formulas:** - Perimeter $P = x + 11 + 5 + 10$ - Area $A = \frac{(\text{base}_1 + \text{base}_2)}{2} \times h$, where $h$ is the height (altitude). 3. **Find the height $h$:** Since the trapezoid has a vertical altitude from the top-left vertex to the bottom base, we can drop a perpendicular from the top-left vertex to the bottom base. Let the foot of the altitude divide the bottom base into two segments: one segment adjacent to the left leg is $d$, and the other segment is $11 - d$. Using the left leg (length $5$) and the altitude $h$, by Pythagoras: $$5^2 = h^2 + d^2 \implies 25 = h^2 + d^2$$ Using the right leg (length $10$) and the altitude $h$, the horizontal distance from the foot of the altitude to the bottom-right vertex is $11 - d - x$ (since top base is $x$ and the foot is under the top-left vertex): $$10^2 = h^2 + (11 - d - x)^2 \implies 100 = h^2 + (11 - d - x)^2$$ 4. **Relate $h^2$ from both equations:** From the first: $$h^2 = 25 - d^2$$ From the second: $$h^2 = 100 - (11 - d - x)^2$$ Set equal: $$25 - d^2 = 100 - (11 - d - x)^2$$ Rearranged: $$(11 - d - x)^2 = 100 - 25 + d^2 = 75 + d^2$$ 5. **Expand and simplify:** $$(11 - d - x)^2 = (11 - x - d)^2 = (11 - x)^2 - 2(11 - x)d + d^2$$ So: $$(11 - x)^2 - 2(11 - x)d + d^2 = 75 + d^2$$ Cancel $d^2$ on both sides: $$(11 - x)^2 - 2(11 - x)d = 75$$ 6. **Solve for $d$:** $$-2(11 - x)d = 75 - (11 - x)^2$$ $$d = \frac{(11 - x)^2 - 75}{2(11 - x)}$$ 7. **Find $h^2$ using $d$:** $$h^2 = 25 - d^2$$ 8. **Area formula:** $$A = \frac{x + 11}{2} \times h = \frac{x + 11}{2} \times \sqrt{25 - d^2}$$ 9. **Perimeter formula:** $$P = x + 11 + 5 + 10 = x + 26$$ **Summary:** - Perimeter: $P = x + 26$ - Area: $$A = \frac{x + 11}{2} \times \sqrt{25 - \left(\frac{(11 - x)^2 - 75}{2(11 - x)}\right)^2}$$ This expresses $A$ and $P$ in terms of $x$.