Answer to Question #200626 in Analytic Geometry for vina

Question #200626

(3.1) Find the area of the triangle with the given vertices A(1, 3), B(−3, 5), and C with C = 2A.

(3.2) Use (3.1) to find the coordinates of the point D such that the quadrilateral ABCD is a parallelogram. 


1
Expert's answer
2021-06-07T06:47:52-0400

(3.1)


"\\overrightarrow{AB}=\\langle-3-1, 5-3\\rangle=\\langle-4, 2\\rangle"


"C=(2(1), 2(3))"


"\\overrightarrow{AC}=\\langle2(1)-1, 2(3)-3\\rangle=\\langle1, 3\\rangle"

"\\overrightarrow{AB}\\times \\overrightarrow{AC}=\\begin{vmatrix}\n \\vec i & \\vec j & \\vec k \\\\\n -4 & 2 & 0 \\\\\n 1 & 3 & 0\n\\end{vmatrix}=(-4(3)-2(1))\\vec k=-14\\vec k"

"=(-4(3)-2(1))\\vec k=-14\\vec k"

"Area_{ABC}=\\dfrac{1}{2}|\\overrightarrow{AB}\\times \\overrightarrow{AC}|=7(units^2)"

Area of the triangle ABC is 7 square units.


(3.2)

Assume that the fourth vertex of parallelogram is "D(x, y)"


"\\overrightarrow{AB}=\\overrightarrow{DC}=\\langle-4, 2\\rangle"

"\\langle 2-x, 6-y\\rangle=\\langle-4, 2\\rangle"

"x=6, y=4"


"D(6, 4)"



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