Given triangle ABC with vertices A(-6, -2), B(2,8), and C(6, -2). Line AB has midpoint D, line BC has midpoint E, and line AC has midpoint F.
Which of the following statements is true?
1. ADEF is only a parallelogram
2. ADEF is a rhombus
3. ADEF is a rectable but not a parallelogram
4. ADEF is not a parallelogram
(1) because while AD || EF, AD=6 and AF = √41
To determine which statement is true, we need to analyze the properties of the quadrilateral ADEF and compare them with the properties of parallelograms, rhombi, rectangles, and other quadrilaterals.
First, let's find the coordinates of the midpoints D, E, and F:
Midpoint D of line AB:
x-coordinate of D = (x-coordinate of A + x-coordinate of B) / 2
= (-6 + 2) / 2
= -4 / 2
= -2
y-coordinate of D = (y-coordinate of A + y-coordinate of B) / 2
= (-2 + 8) / 2
= 6 / 2
= 3
So, midpoint D is (-2, 3).
Using the same process, we find the coordinates of midpoints E and F:
Midpoint E of line BC:
x-coordinate of E = (x-coordinate of B + x-coordinate of C) / 2
= (2 + 6) / 2
= 8 / 2
= 4
y-coordinate of E = (y-coordinate of B + y-coordinate of C) / 2
= (8 + -2) / 2
= 6 / 2
= 3
So, midpoint E is (4, 3).
Midpoint F of line AC:
x-coordinate of F = (x-coordinate of A + x-coordinate of C) / 2
= (-6 + 6) / 2
= 0 / 2
= 0
y-coordinate of F = (y-coordinate of A + y-coordinate of C) / 2
= (-2 + -2) / 2
= -4 / 2
= -2
So, midpoint F is (0, -2).
Now that we have the coordinates of A, D, E, and F, we can determine the nature of quadrilateral ADEF by analyzing its sides and angles.
1. Parallelogram: A quadrilateral is a parallelogram if opposite sides are parallel.
To determine if the opposite sides of ADEF are parallel, we can calculate the slopes of the sides AD, DE, EF, and FA:
Slope of AD = (y-coordinate of D - y-coordinate of A) / (x-coordinate of D - x-coordinate of A)
= (3 - (-2)) / (-2 - (-6))
= 5 / 4
Slope of DE = (y-coordinate of E - y-coordinate of D) / (x-coordinate of E - x-coordinate of D)
= (3 - 3) / (4 - (-2))
= 0 / 6
= 0
Slope of EF = (y-coordinate of F - y-coordinate of E) / (x-coordinate of F - x-coordinate of E)
= (-2 - 3) / (0 - 4)
= -5 / (-4)
= 5 / 4
Slope of FA = (y-coordinate of A - y-coordinate of F) / (x-coordinate of A - x-coordinate of F)
= (-2 - (-2)) / (-6 - 0)
= 0 / -6
= 0
Since the slopes of opposite sides AD and EF, as well as DE and FA, are equal, we can conclude that ADEF is a parallelogram.
2. Rhombus: A parallelogram is a rhombus if all four sides are congruent.
To determine if ADEF is a rhombus, we need to calculate the lengths of the sides AD, DE, EF, and FA and check if they are equal.
Length of AD = sqrt((x-coordinate of D - x-coordinate of A)^2 + (y-coordinate of D - y-coordinate of A)^2)
= sqrt((-2 - (-6))^2 + (3 - (-2))^2)
= sqrt(4^2 + 5^2)
= sqrt(16 + 25)
= sqrt(41)
Length of DE = sqrt((x-coordinate of E - x-coordinate of D)^2 + (y-coordinate of E - y-coordinate of D)^2)
= sqrt((4 - (-2))^2 + (3 - 3)^2)
= sqrt(6^2 + 0^2)
= sqrt(36 + 0)
= sqrt(36)
= 6
Length of EF = sqrt((x-coordinate of F - x-coordinate of E)^2 + (y-coordinate of F - y-coordinate of E)^2)
= sqrt((0 - 4)^2 + (-2 - 3)^2)
= sqrt((-4)^2 + (-5)^2)
= sqrt(16 + 25)
= sqrt(41)
Length of FA = sqrt((x-coordinate of A - x-coordinate of F)^2 + (y-coordinate of A - y-coordinate of F)^2)
= sqrt((-6 - 0)^2 + (-2 - (-2))^2)
= sqrt((-6)^2 + 0^2)
= sqrt(36 + 0)
= sqrt(36)
= 6
Since the lengths of all four sides are not equal (AD ≠ DE ≠ EF ≠ FA), we can conclude that ADEF is not a rhombus.
3. Rectangle: A parallelogram is a rectangle if all four angles are right angles (each measuring 90 degrees).
To determine if ADEF is a rectangle, we need to calculate the slopes of the sides AD, DE, EF, and FA and check if they are perpendicular (have negative reciprocal slopes).
Slope of AD = 5/4
Slope of DE = 0 (horizontal line)
Slope of EF = 5/4
Slope of FA = 0 (horizontal line)
Since the slopes of the non-horizontal sides AD and EF are equal, we can conclude that they are parallel. However, the condition for perpendicularity is not satisfied since the slope of DE is 0 (horizontal line) instead of having a negative reciprocal slope.
Therefore, ADEF is not a rectangle.
4. Parallelogram: We have already established that ADEF is a parallelogram (opposite sides are parallel). Therefore, statement 4 is true.
To summarize:
1. False (ADEF is a parallelogram, not just a parallelogram).
2. False (ADEF is not a rhombus).
3. False (ADEF is not a rectangle).
4. True (ADEF is a parallelogram).
The correct statement is: ADEF is not a parallelogram (option 4).