Analyzing Quadrilaterals | Comparing Sides, Angles, and Areas

The vertices of a quadrilateral ABCD are A(1, −3), B(4, −3), C(4, −5), and D(−1, −5). The vertices of another quadrilateral EFCD are E(1, −7), F(4, −7), C(4, −5), and D(−1, −5). Which conclusion is true about the quadrilaterals? The ratio of their corresponding sides is not equal.The angles and sides overlap when one quadrilateral is placed on the other.The measures of the corresponding angles are different.The shape of the quadrilaterals is same but their areas are different.

To determine which conclusion is true about the quadrilaterals, we need to analyze the given information

To determine which conclusion is true about the quadrilaterals, we need to analyze the given information.

1. The ratio of their corresponding sides is not equal:
By examining the coordinates of the vertices, we can find the lengths of the sides of both quadrilaterals. Let’s calculate the lengths of the sides of both quadrilaterals:

Quadrilateral ABCD:
Side AB: √[(4-1)^2 + (-3-(-3))^2] = √[3^2 + 0^2] = √9 = 3
Side BC: √[(4-4)^2 + (-5-(-3))^2] = √[0^2 + (-2)^2] = √4 = 2
Side CD: √[(-1-4)^2 + (-5-(-5))^2] = √[(-5)^2 + 0^2] = √25 = 5
Side DA: √[(1-(-1))^2 + (-3-(-5))^2] = √[2^2 + 2^2] = √8 ≈ 2.828

Quadrilateral EFCD:
Side EF: √[(4-1)^2 + (-7-(-7))^2] = √[3^2 + 0^2] = √9 = 3
Side FC: √[(4-4)^2 + (-5-(-7))^2] = √[0^2 + 2^2] = √4 = 2
Side CD: √[(-1-4)^2 + (-5-(-5))^2] = √[(-5)^2 + 0^2] = √25 = 5
Side DE: √[(1-(-1))^2 + (-7-(-5))^2] = √[2^2 + 2^2] = √8 ≈ 2.828

Comparing the lengths of the corresponding sides:

AB/EF = 3/3 = 1
BC/FC = 2/2 = 1
CD/CD = 5/5 = 1
DA/DE ≈ 2.828/2.828 = 1

Hence, the ratio of the corresponding sides of the two quadrilaterals is indeed equal. Therefore, we can conclude that “The ratio of their corresponding sides is equal” is false.

2. The angles and sides overlap when one quadrilateral is placed on the other:
To determine if the angles and sides overlap, we can compare the coordinates of the common points of each quadrilateral:

Quadrilateral ABCD and Quadrilateral EFCD have the common points C(4, -5) and D(-1, -5).

Since the common points coincide, we can conclude that “The angles and sides overlap when one quadrilateral is placed on the other” is true.

3. The measures of the corresponding angles are different:
To determine if the measures of the corresponding angles are different, we need to calculate the angles between the sides.

Quadrilateral ABCD:
Angle ABC: acos((3^2 + 2^2 – 5^2)/(2*3*2)) = acos((-4)/12) ≈ 2.094 radians ≈ 120 degrees
Angle BCD: acos((2^2 + 5^2 – 3^2)/(2*2*5))= acos((18)/20) ≈ 0.722 radians ≈ 41.4 degrees
Angle CDA: acos((5^2 + 2^2 – 3^2)/(2*5*2)) = acos((14)/20) ≈ 1.154 radians ≈ 66.2 degrees
Angle DAB: π – (Angle ABC + Angle BCD + Angle CDA) ≈ π – (120 + 41.4 + 66.2) ≈ 207.4 degrees

Quadrilateral EFCD:
Angle EFC: acos((3^2 + 2^2 – 5^2)/(2*3*2)) = acos((-4)/12) ≈ 2.094 radians ≈ 120 degrees
Angle FCD: acos((2^2 + 5^2 – 3^2)/(2*2*5))= acos((18)/20) ≈ 0.722 radians ≈ 41.4 degrees
Angle CDE: acos((5^2 + 2^2 – 3^2)/(2*5*2)) = acos((14)/20) ≈ 1.154 radians ≈ 66.2 degrees
Angle DEF: π – (Angle EFC + Angle FCD + Angle CDE) ≈ π – (120 + 41.4 + 66.2) ≈ 207.4 degrees

Comparing the measures of the corresponding angles:

Angle ABC ≈ Angle EFC ≈ 120 degrees
Angle BCD ≈ Angle FCD ≈ 41.4 degrees
Angle CDA ≈ Angle CDE ≈ 66.2 degrees
Angle DAB ≈ Angle DEF ≈ 207.4 degrees

Hence, the measures of the corresponding angles are the same, and “The measures of the corresponding angles are different” is false.

4. The shape of the quadrilaterals is the same, but their areas are different:
By comparing the corresponding sides and angles, we have already determined that the shape of the quadrilaterals is the same. However, to conclude that their areas are different, we need to calculate the areas of both quadrilaterals.

Using the Shoelace Formula or the Heron’s Formula, we can calculate the areas:

Quadrilateral ABCD:
Area ABCD = |(1*(-3) + 4*(-5) + (-1)*(-5) + 4*(-3))/2| = 8 square units

Quadrilateral EFCD:
Area EFCD = |(1*(-7) + 4*(-5) + (-1)*(-5) + 4*(-7))/2| = 6 square units

Hence, the areas of the quadrilaterals are different, and “The shape of the quadrilaterals is the same, but their areas are different” is true.

In conclusion, the correct statement is: “The shape of the quadrilaterals is the same, but their areas are different.”

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