Question

what is the most precise name for a quadrilateral ABCD with vertices A(-5,2),B(-3,5), C(4,5) and D(2,2)

Answer 1

Answer:

The most precise name for a quadrilateral ABCD is a parallelogram

Step-by-step explanation:

step 1

Draw the figure

we have that

The coordinates of a quadrilateral are

A(-5,2),B(-3,5), C(4,5) and D(2,2)

using a graphing tool

Plot the coordinates to better understand the problem

see the attached figure

step 2

Find the length sides of the quadrilateral

the formula to calculate the distance between two points is equal to

$d=\sqrt{(y2-y1)^{2}+(x2-x1)^{2}}$

Find the length side AB

A(-5,2),B(-3,5)

substitute in the formula

$d=\sqrt{(5-2)^{2}+(-3+5)^{2}}$

$d=\sqrt{(3)^{2}+(2)^{2}}$

$d_A_B=\sqrt{13}\ units$

Find the length side CD

C(4,5), D(2,2)

substitute in the formula

$d=\sqrt{(2-5)^{2}+(2-4)^{2}}$

$d=\sqrt{(-3)^{2}+(-2)^{2}}$

$d_C_D=\sqrt{13}\ units$

Find the length side AD

A(-5,2), D(2,2)

substitute in the formula

$d=\sqrt{(2-2)^{2}+(2+5)^{2}}$

$d=\sqrt{(0)^{2}+(7)^{2}}$

$d_A_D=7\ units$

Find the length side BC

B(-3,5), C(4,5)

substitute in the formula

$d=\sqrt{(5-5)^{2}+(4+3)^{2}}$

$d=\sqrt{(0)^{2}+(7)^{2}}$

$d_B_C=7\ units$

step 3

Find the slope of the length sides of the quadrilateral

The formula to calculate the slope between two points is equal to

$m=\frac{y2-y1}{x2-x1}$

Find the slope of the length side AB

A(-5,2),B(-3,5)

substitute in the formula

$m=\frac{5-2}{-3+5}$

$m_A_B=\frac{3}{2}$

Find the slope of the length side CD

C(4,5), D(2,2)

substitute in the formula

$m=\frac{2-5}{2-4}$

$m_C_D=\frac{3}{2}$

Find the slope of the length side AD

A(-5,2), D(2,2)

substitute in the formula

$m=\frac{2-2}{2+5}$

$m_A_D=0$  ----> is a horizontal line

Find the slope of the length side BC

B(-3,5), C(4,5)

substitute in the formula

substitute in the formula

$m=\frac{5-5}{4+3}$

$m_B_C=0$  ----> is a horizontal line

step 4

Compare the length sides

$d_A_B=\sqrt{13}\ units$

$d_C_D=\sqrt{13}\ units$

$d_A_D=7\ units$

$d_B_C=7\ units$

opposite sides are congruent

step 5

Compare the slopes

Remember that

If the lines are parallel, then their slopes are the same

$m_A_B=\frac{3}{2}$

$m_C_D=\frac{3}{2}$

$m_A_D=0$

$m_B_C=0$

so

opposite sides are parallel

therefore

The most precise name for a quadrilateral ABCD is a parallelogram