Respuesta :
Answer:
Numbering the options, we have;
1) Side B'A' has a slope of −1 and is perpendicular to side BA.
2) Side B'A' has a slope of 1 and is parallel to side BA.
3) Side B'A' has a slope of 1 and is perpendicular to side BA.
4) Side B'A' has a slope of −1 and is parallel to side BA.
The correct option is;
1) Side B'A' has a slope of -1 and is perpendicular to side BA
Step-by-step explanation:
The given coordinates are;
A(3, 5) B(1, 3), C(5, -1) and D(7, 1)
The slope of BA is found as follows;
[tex]Slope \, of BA= \dfrac{y_{2}-y_{1}}{x_{2}-x_{1}} =\dfrac{3-5}{1-3}= \dfrac{-2}{-2} = 1[/tex]
Rotation of a line through 90 degrees gives
(x, y) will be (y, -x)
Therefore, the coordinates of A' = (5, -3)
The coordinates of B' = (3, -1)
Then the slope is given as follows;
[tex]Slope \, of \, B'A' =\dfrac{-1 -(-3)}{3-5}= \dfrac{2}{-2} = -1[/tex]
Therefore side B'A' has a slope of -1 and is perpendicular to side BA.
Option (4) will be the correct option.
Coordinates of the vertices of rectangle ABCD are,
- A(3, 5), B(1, 3), C(5, -1) and D()
Slope of a segment with endpoints [tex](x_1,y_1)[/tex] and [tex](x_2,y_2)[/tex] is given by the expression,
Slope 'm' = [tex]\frac{y_2-y_1}{x_2-x_1}[/tex]
Therefore, slope of segment BA (m₁) = [tex]\frac{3-5}{1-3}[/tex] = 1
If a point (h, k) is rotated 90° clockwise about the origin, rule to be followed,
(h, k) → (-k, h)
Following this rule,
B(1, 3) → B'(-3, 1)
A(3, 5)→ A'(-5, 3)
Slope of segment B'A' (m₂) = [tex]\frac{1-3}{-3+5}=-1[/tex]
Expression representing two segments with slopes [tex]m_1[/tex] and [tex]m_2[/tex] to be perpendicular,
[tex]m_1\times m_2=-1[/tex]
Since, slopes of two segments BA and B'A' are (1) and (-1).
(1) × (-1) = -1
Therefore, both the segments BA and B'A' will be perpendicular.
Option (4) will be the answer.
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