1. **Problem 18:** Classify the polygon with vertices $S(7, 2)$, $T(1, 2)$, $U(3, -3)$, and $V(5, -3)$ and find its area. 2. **Step 1: Identify the polygon type.** The polygon has 4 vertices, so it is a quadrilateral. 3. **Step 2: Check if it is a trapezoid or parallelogram by comparing slopes of opposite sides.** Calculate slopes: - $ST = \frac{2-2}{1-7} = \frac{0}{-6} = 0$ - $UV = \frac{-3 - (-3)}{5-3} = \frac{0}{2} = 0$ - $TU = \frac{-3-2}{3-1} = \frac{-5}{2} = -2.5$ - $SV = \frac{-3-2}{5-7} = \frac{-5}{-2} = 2.5$ Since $ST$ is parallel to $UV$ (both slope 0) and $TU$ is parallel to $SV$ (slopes $-2.5$ and $2.5$ are not equal), the polygon is a trapezoid. 4. **Step 3: Use the Shoelace formula to find the area of the polygon.** The Shoelace formula for vertices $(x_1,y_1), (x_2,y_2), ..., (x_n,y_n)$ is: $$\text{Area} = \frac{1}{2} \left| \sum_{i=1}^{n-1} (x_i y_{i+1}) + x_n y_1 - \sum_{i=1}^{n-1} (y_i x_{i+1}) - y_n x_1 \right|$$ 5. **Step 4: Apply the formula:** Vertices in order: $S(7,2), T(1,2), U(3,-3), V(5,-3)$ Calculate sums: $$\sum x_i y_{i+1} = 7 \times 2 + 1 \times (-3) + 3 \times (-3) + 5 \times 2 = 14 - 3 - 9 + 10 = 12$$ $$\sum y_i x_{i+1} = 2 \times 1 + 2 \times 3 + (-3) \times 5 + (-3) \times 7 = 2 + 6 - 15 - 21 = -28$$ 6. **Step 5: Calculate area:** $$\text{Area} = \frac{1}{2} |12 - (-28)| = \frac{1}{2} \times 40 = 20$$ 7. **Answer for Problem 18:** The polygon is a trapezoid and its area is $20$ square units. --- 1. **Problem 19:** In isosceles trapezoid $TUVW$, find $UW$ if $TX = 23$ inches and $VX = 28.7$ inches, where $X$ is the intersection of diagonals. 2. **Step 1: Recall property of isosceles trapezoid diagonals.** In an isosceles trapezoid, the diagonals are equal in length and intersect each other such that the segments are proportional. 3. **Step 2: Use the property that the diagonals are divided proportionally by their intersection point $X$.** Given $TX = 23$ and $VX = 28.7$, the full diagonal $TW = TX + VX = 23 + 28.7 = 51.7$ inches. 4. **Step 3: Since trapezoid is isosceles, the other diagonal $UV$ is also $51.7$ inches.** 5. **Step 4: Use the triangle formed by points $U$, $W$, and $X$ to find $UW$.** Since $X$ is the intersection of diagonals, $UW$ is the base of the trapezoid. 6. **Step 5: Use the fact that in an isosceles trapezoid, the diagonals bisect each other proportionally, so $UW = TX + VX = 51.7$ inches.** 7. **Answer for Problem 19:** $UW = 51.7$ inches.