in the coordinate plane, points a, b, and c have coordinates (1,2), (4,2), and (4, - 1) respectively.\n1…

in 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.

in 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

Answer:

  1. Right - angled triangle
  2. (4.5)
  3. ((1,-1))

Explanation:

Step1: Determine the shape of (\triangle ABC)

  • Calculate the lengths of the sides using the distance formula (d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}).
  • For (AB): (x_1 = 1,y_1 = 2,x_2 = 4,y_2 = 2), (AB=\sqrt{(4 - 1)^2+(2 - 2)^2}=\sqrt{9+0}=3).
  • For (BC): (x_1 = 4,y_1 = 2,x_2 = 4,y_2=-1), (BC=\sqrt{(4 - 4)^2+(-1 - 2)^2}=\sqrt{0 + 9}=3).
  • For (AC): (x_1 = 1,y_1 = 2,x_2 = 4,y_2=-1), (AC=\sqrt{(4 - 1)^2+(-1 - 2)^2}=\sqrt{9 + 9}=\sqrt{18}=3\sqrt{2}).
  • Since (AB^{2}+BC^{2}=3^{2}+3^{2}=9 + 9 = 18) and (AC^{2}=(3\sqrt{2})^{2}=18), by the Pythagorean theorem ((a^{2}+b^{2}=c^{2})), (\triangle ABC) is a right - angled triangle with (\angle B = 90^{\circ}).

Step2: Calculate the area of (\triangle ABC)

  • The formula for the area of a right - angled triangle is (A=\frac{1}{2}\times base\times height).
  • Here, (base = AB = 3) and (height=BC = 3).
  • So, (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 (AB) is parallel to (CD) and (AD) is parallel to (BC).
  • The (x) - coordinate of (D) is the same as the (x) - coordinate of (A) (because (AD\parallel BC) and (BC) is a vertical line), and the (y) - coordinate of (D) is the same as the (y) - coordinate of (C) (because (CD\parallel AB) and (AB) is a horizontal line).
  • So, the coordinates of (D) are ((1,-1)).
  • Check the lengths:
    • (AD=\sqrt{(1 - 1)^2+(-1 - 2)^2}=3), (CD=\sqrt{(4 - 1)^2+(-1+1)^2}=3).
    • (AB = 3), (BC = 3). Also, (AB\parallel CD) (both have a slope of (m = 0)) and (AD\parallel BC) (both have an undefined slope).