Can you prove ABCD is a parallelogram based on the given information? Explain.
Given: x = 5, y = 4
Prove: ABCD is a parallelogram.
The image consists of a quadrilatera ABCD with their sides AB, BC, CD, DA as 3x + 12, 7y - 8, 6x - 3 and 5y respectively.
To be a ||gm , opposite sides must be equal
7y-8 = 5y and 6x-3 = 3x+12
7y - 8 = 5y
2y = 8
y = 4
6x-3 = 3x+12
3x = 15
x = 5
Since the given information about x and y match those solutions, yes, it is a parallogram
or
just sub in the given values into the 4 different expressions to see that indeed opposite sides are equal
To prove that ABCD is a parallelogram based on the given information, we need to show that opposite sides are parallel.
Given that the sides of ABCD are AB = 3x + 12, BC = 7y - 8, CD = 6x - 3, and DA = 5y, we can start by comparing the slopes of opposite sides.
The slope of a line can be determined by examining the coefficients of x and y in the equation of the line. In general, if two lines have the same slope, they are parallel.
Let's find the slopes of the opposite sides:
1. For side AB: The equation of AB is 3x + 12. The coefficient of x is 3, which represents the slope.
So, the slope of AB is 3.
2. For side CD: The equation of CD is 6x - 3. The coefficient of x is 6, which represents the slope.
So, the slope of CD is 6.
Next, let's compare the slopes of AB and CD. We can see that the slopes are different (3 and 6), which means that AB and CD are not parallel.
Therefore, based on the given information, we cannot prove that ABCD is a parallelogram.
To prove that ABCD is a parallelogram, we need to show that opposite sides are parallel.
Let's first find the equations for the slopes of each side of ABCD using the given information.
The slope of a line is given by the formula: slope = (change in y) / (change in x).
For side AB:
The equation for AB is 3x + 12.
To find the slope, we take the coefficient of x, which is 3.
So the slope of AB is 3.
For side BC:
The equation for BC is 7y - 8.
To find the slope, we take the coefficient of y, which is 7.
So the slope of BC is 7.
For side CD:
The equation for CD is 6x - 3.
To find the slope, we take the coefficient of x, which is 6.
So the slope of CD is 6.
For side DA:
The equation for DA is 5y.
To find the slope, we take the coefficient of y, which is 5.
So the slope of DA is 5.
Now, to prove that ABCD is a parallelogram, we need to show that opposite sides have equal slopes.
The slopes we found are: AB = 3, BC = 7, CD = 6, and DA = 5.
Opposite sides AB and CD have equal slopes: 3 = 6.
Opposite sides BC and DA have equal slopes: 7 = 5.
Since opposite sides have equal slopes, we can conclude that ABCD is a parallelogram based on the given information.