Question

What is the most precise term for quadrilateral ABCD with vertices A(3,1) , B(4,5) , C(7,5) , D(7,2) answer choices; A. KITE B. SQUARE C. RHOMBUS D. PARALLELOGRAM

Answer 1

Answer:

Kite

Step-by-step explanation:

We are given that a quadrilateral ABCD with vertices A(3,1),B(4,5),C(7,5) and D(7,2).

We have to find the type of quadrilateral.

Construction: Join A to C and B to D.

Mid- point formula

$x=\frac{x_1+x_2}{2},y=\frac{y_1+y_2}{2}$

Using this formula

Mid point of diagonal AC,$x=\frac{3+7}{2}=5, y=\frac{1+5}{2}=3$=(5,3)

Mid- point of diagonal of BD=$x=\frac{4+7}{2}=\frac{11}{2}, y=\frac{5+2}{2}=\frac{7}{2}=(\frac{11}{2},\frac{7}{2})$

We know that diagonals of parallelogram , square and rhombus are bisect to each other.

Therefore, mid-points of both diagonal are same .

Slope formula =$\frac{y_2-y_1}{x_2-x_1}$

When two lines are perpendicular then product of their slopes=-1

Slope of AC=$\frac{5-1}{7-3}=1$

Slope of BD=$\frac{2-5}{7-4}=-1$

Product of slope of AC and  slope of BD=-1

Hence, diagonal AC and diagonal BD are perpendicular to each other.

But there are mid - point of both diagonal are not same.

When the perpendicular cut to each other and do not bisect to each other then the quadrilateral ABCD is kite.

Hence, the given quadrilateral is kite.

Answer: Kite

Answer 2

Answer:

Step-by-step explanation:

Given are the vertices of a quadrilateral.

WE have to find the name of the quadrilateral

WE have slope of AC = (5-1)/(7-3) = 1

and slope of BD = (2-5)/(7-4) = -1

Since m1m2 =-1 we get

the diagonals cut at right angles

This property is special property for square, rhombus and kite only.

Let us check if diagonals bisect each other

Mid point of AC = $(\frac{3+7}{2} ,\frac{1+5}{2} =(5,3)$

Mid point of BD =$(\frac{4+7}{2} ,\frac{2+5}{2} =\neq 5,3)$

Since diagonals do not bisect this is neither rhombus nor square.

Only possibility is kite