Question

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

1. plot points a, b, and c on a coordinate grid, then connect them to form a figure. what is the shape of this figure?
2. calculate the area of the figure formed by points a, b, and c.
3. 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.

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).

Answer:

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