find the perimeter of the polygon with the given vertices.\n3. d(-3, 2), e(4, 2), f(4, -3)\n4. g(-3, 2)…
find the perimeter of the polygon with the given vertices.\n3. d(-3, 2), e(4, 2), f(4, -3)\n4. g(-3, 2), h(2, 2), j(-1, -3)\n5. k(-1, 1), l(4, 1), m(2, -2), n(-3, -2)\n6. q(-4, -1), r(1, 4), s(4, 1), t(-1, -4)\nfind the area of the polygon with the given vertices.\n7. g(2, 2), h(3, -1), j(-2, -1)\n8. n(-1, 1), p(2, 1), q(2, -2), r(-1, -2)\n9. f(-2, 3), g(1, 3), h(1, -1), j(-2, -1)\n10. k(-3, 3), l(3, 3), m(3, -1), n(-3, -1)
Answer
- For problem 3 (finding perimeter of polygon with vertices (D(-3,2)), (E(4,2)), (F(4, - 3))):
- First, find the lengths of the sides using the distance - formula (d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}). For horizontal or vertical lines, the distance is simply the difference in the non - equal coordinates.
- Step 1: Find the length of (DE)
- Since (D(-3,2)) and (E(4,2)) have the same (y) - coordinate ((y = 2)), the length of (DE) is (|4-(-3)|=|4 + 3|=7).
- Step 2: Find the length of (EF)
- Since (E(4,2)) and (F(4,-3)) have the same (x) - coordinate ((x = 4)), the length of (EF) is (|2-(-3)|=|2 + 3|=5).
- Step 3: Find the length of (DF)
- Using the distance formula (d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}), where (x_1=-3,y_1 = 2,x_2 = 4,y_2=-3).
- (d=\sqrt{(4-(-3))^2+(-3 - 2)^2}=\sqrt{(7)^2+(-5)^2}=\sqrt{49 + 25}=\sqrt{74}\approx8.6).
- Step 4: Calculate the perimeter (P)
- (P=DE + EF+DF=7 + 5+\sqrt{74}=12+\sqrt{74}\approx12 + 8.6=20.6).
- Step 1: Find the length of (DE)
- First, find the lengths of the sides using the distance - formula (d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}). For horizontal or vertical lines, the distance is simply the difference in the non - equal coordinates.
- For problem 7 (finding area of polygon with vertices (G(2,2)), (H(3,-1)), (J(-2,-1))):
- We can use the Shoelace formula for the area of a polygon with vertices ((x_1,y_1),(x_2,y_2),\cdots,(x_n,y_n)) which is (A=\frac{1}{2}\left|\sum_{i = 1}^{n - 1}x_iy_{i+1}-x_{i + 1}y_i\right|+\frac{1}{2}\left|x_ny_1-x_1y_n\right|).
- Here (n = 3), (x_1 = 2,y_1 = 2,x_2 = 3,y_2=-1,x_3=-2,y_3=-1).
- Step 1: Calculate (\sum_{i = 1}^{2}(x_iy_{i+1}-x_{i + 1}y_i)+(x_3y_1-x_1y_3))
- ((x_1y_2-x_2y_1)+(x_2y_3-x_3y_2)+(x_3y_1-x_1y_3))
- ((2\times(-1)-3\times2)+(3\times(-1)-(-2)\times(-1))+((-2)\times2-2\times(-1)))
- ((-2 - 6)+(-3 - 2)+(-4 + 2))
- (-8-5 - 2=-15).
- Step 2: Calculate the area (A)
- (A=\frac{1}{2}|-15| = 7.5).
- We can use the Shoelace formula for the area of a polygon with vertices ((x_1,y_1),(x_2,y_2),\cdots,(x_n,y_n)) which is (A=\frac{1}{2}\left|\sum_{i = 1}^{n - 1}x_iy_{i+1}-x_{i + 1}y_i\right|+\frac{1}{2}\left|x_ny_1-x_1y_n\right|).
If we consider just one of the problems, for example, problem 3:
Explanation:
Step1: Find length of (DE)
Since (y) - coordinates are same, (DE=|4-(-3)| = 7)
Step2: Find length of (EF)
Since (x) - coordinates are same, (EF=|2-(-3)| = 5)
Step3: Find length of (DF)
Use distance formula (DF=\sqrt{(4-(-3))^2+(-3 - 2)^2}=\sqrt{74})
Step4: Calculate perimeter
(P=DE + EF+DF=7 + 5+\sqrt{74}=12+\sqrt{74})
Answer:
(12+\sqrt{74})