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2 answers:
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Answer:
a. The four sides of the quadrilateral ABCD are equal, therefore, ABCD is a rhombus
b. The equation of the diagonal line AC is y = 5 - x
The equation of the diagonal line BD is y = 5 - x
c. The diagonal lines AC and BD of the quadrilateral ABCD are perpendicular to each other
Step-by-step explanation:
The vertices of the given quadrilateral are;
A(1, 4), B(6, 6), C(4, 1) and D(-1, -1)
a. The length, l, of the sides of the given quadrilateral are given as follows;
The length of side AB, with A = (1, 4) and B = (6, 6) gives;
The length of side BC, with B = (6, 6) and C = (4, 1) gives;
The length of side CD, with C = (4, 1) and D = (-1, -1) gives;
The length of side DA, with D = (-1, -1) and A = (1,4) gives;
Therefore, each of the lengths of the sides of the quadrilateral ABCD are equal to √(29), and the quadrilateral ABCD is a rhombus
b. The diagonals are AC and BD
The slope, m, of AC is given by the formula for the slope of a straight line as follows;
Therefore;
The equation of the diagonal AC in point and slope form is given as follows;
y - 4 = -1×(x - 1)
y = -x + 1 + 4
The equation of the diagonal AC is y = 5 - x
The equation of the diagonal BD in point and slope form is given as follows;
y - 6 = 1×(x - 6)
y = x - 6 + 6 = x
The equation of the diagonal BD is y = x
c. Comparing the lines AC and BD with equations, y = 5 - x and y = x, which are straight line equations of the form y = m·x + c, where m = the slope and c = the x intercept, we have;
The slope m for the diagonal AC = -1 and the slope m for the diagonal BD = 1, therefore, the slopes are opposite signs
The point of intersection of the two diagonals is given as follows;
5 - x = x
∴ x = 5/2 = 2.5
y = x = 2.5
The lines intersect at (2.5, 2.5), given that the slopes, m₁ = -1 and m₂ = 1 of the diagonals lines satisfy the condition for perpendicular lines m₁ = -1/m₂, therefore, the diagonals are perpendicular.
The slopes of perpendicular lines are opposite reciprocals
The true statement is that segments FG and HJ are perpendicular
<h3>How to determine the relationship between the segments</h3>The coordinates of the points are given as:
F = (3,1)
G = (5,2)
H = (2,4)
J = (1,6)
Start by calculating the slopes of FG and HJ using the following slope formula
So, we have:
Also, we have:
To determine the relationship, we make use of the following highlights
- Parallel lines have the same slope
- The slopes of perpendicular lines are opposite reciprocals
From the computation above, we have:
- The slopes of both lines are not equal
- The slopes are opposite reciprocals i.e. 2 = -1(-1/2)
Hence, segment FG and HJ are perpendicular
Read more about perpendicular lines at: