Respuesta :
Answer:
The given quadrilateral is a rectangle.
Step-by-step explanation:
We are given the vertices of a quadrilateral,
We can use the slopes and modulus of all sides to determine whether it is a rectangle or not. The four sides of the quadrilateral are Ab, BC, CD and AD. So finding slopes of all sides using the formula
[tex]Slope=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}[/tex]
where x_1 and y_1 are the coordinates of firsst vertex and x_2 and y_2 are coordinates of second vertex.
So.
Slopes are:
Slope of AB = 2 and Length=2.23
Slope of BC = -1/2 and length = 4.47
Slope of CD = 2 and length = 2.23
Slope of AD = -1/2 and length = 4.47
As we can see that the slope of AB and CD is equal which means that they are parallel and their lengths are also same.
Similarly BC and AD have same slope which means that they are also parallel and their lengths are also equal.
More over the product of slopes of AB and CD, BC and CD, CD and AD and AD and AB is equal to -1 which indicated that these sides are parallel to each other so it can be concluded that as the opposite sides of the quadrilateral are equal and the interior angles are 90 degrees so the quadrilateral is a rectangle ..
Answer:
I am given that the vertices of quadrilateral ABCD are A(-1, 6), B(-2, 4), C(2, 2), and D(3, 4). The slope formula applied to each pair of adjacent vertices gives the slopes (m) of the sides:
slope of AB (m AB) = 4 - 6 / -2 - -1) = -2 / -1 =2
slope of BC (m BC) = 2 - 4 / 2 - (-2) = -2 / 4 = - 1 / 2
slope of CD (m CD) = 4 - 2 / 3 -2 = 2 / 1 = 2
slope of DA (m DA) = 6 - 4 / - 1 -3 = 2 / - 4 = - 1 / 2
Because line segments with equal slopes are parallel, segment AB is parallel to segment CD and segment BC is parallel to segment DA.
Multiplying the slopes of one pair of adjacent sides, I find that.
mAB * mBC = 2 * - 1/2 = - 1
If the product of the slopes of two segments is -1, then they are perpendicular. Since both pairs of opposite sides have been proven parallel, proving one pair of adjacent sides perpendicular implies that the other pair of adjacent sides are also perpendicular. Therefore, by definition, quadrilateral ABCD is a rectangle.
Step-by-step explanation:
