question:\nin the coordinate plane, points a, b, and c have coordinates (1,2), (4,2), and (4, - 1)…

question:\nin the coordinate plane, points a, b, and c have coordinates (1,2), (4,2), and (4, - 1) respectively.\n1. plot points a, b, and c on a coordinate grid, then connect them to form a figure. what is the shape of this figure?\n2. calculate the area of the figure formed by points a, b, and c.\n3. find the coordinates of point d such that quadrilateral abcd is a rectangle. plot point d and verify the rectangle by checking the properties of its sides.

question:\nin the coordinate plane, points a, b, and c have coordinates (1,2), (4,2), and (4, - 1) respectively.\n1. plot points a, b, and c on a coordinate grid, then connect them to form a figure. what is the shape of this figure?\n2. calculate the area of the figure formed by points a, b, and c.\n3. find the coordinates of point d such that quadrilateral abcd is a rectangle. plot point d and verify the rectangle by checking the properties of its sides.

Answer

Explanation:

Step1: Determine the shape of the figure

  • Calculate the lengths of (AB) and (BC).
    • The distance formula between two points ((x_1,y_1)) and ((x_2,y_2)) is (d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}).
    • For (A(1,2)) and (B(4,2)): (AB=\sqrt{(4 - 1)^2+(2 - 2)^2}=\sqrt{3^2+0^2}=3).
    • For (B(4,2)) and (C(4,-1)): (BC=\sqrt{(4 - 4)^2+(-1 - 2)^2}=\sqrt{0^2+(-3)^2}=3).
    • Since (AB) is horizontal ((y) - coordinate is same) and (BC) is vertical ((x) - coordinate is same), the figure formed by (A), (B), (C) is a right - triangle.

Step2: Calculate the area of the right - triangle

  • The formula for the area of a right - triangle is (A=\frac{1}{2}\times base\times height).
  • Here, base (AB = 3) and height (BC=3).
  • (A=\frac{1}{2}\times3\times3=\frac{9}{2}=4.5).

Step3: Find the coordinates of point (D)

  • In a rectangle, opposite sides are equal and parallel.
  • If (A(1,2)), (B(4,2)), (C(4,-1)), for (ABCD) to be a rectangle.
  • The (x) - coordinate of (D) is the same as the (x) - coordinate of (A) (since (AD\parallel BC)) and the (y) - coordinate of (D) is the same as the (y) - coordinate of (C) (since (CD\parallel AB)).
  • So, (D(1,-1)).
    • Check the lengths:
      • (AD=\sqrt{(1 - 1)^2+(-1 - 2)^2}=3), (CD=\sqrt{(1 - 4)^2+(-1+1)^2}=3).
      • (AB = 3), (BC = 3), (AD = 3), (CD = 3) and (AB\parallel CD), (BC\parallel AD)

Answer:

  1. The shape of the figure formed by (A), (B), (C) is a right - triangle.
  2. The area of the figure is (4.5).
  3. The coordinates of point (D) is ((1,-1)).