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The figure is quadrilateral L M N O. It appears to be a parallelogram.
If ON = 8x − 8,
LM = 7x + 4, NM = x − 5,
and OL = 3y − 6,
find the values of x and y for which LMNO must be a parallelogram. The diagram is not drawn to scale.
To determine the values of x and y for which quadrilateral LMNO must be a parallelogram, we need to use the properties of parallelograms.
One property of parallelograms is that opposite sides are congruent. In quadrilateral LMNO, the opposite sides are LM and NO.
Given: LM = 7x + 4 and NO = ON = 8x - 8
To make LMNO a parallelogram, we set LM equal to NO:
7x + 4 = 8x - 8
Now, we can solve this equation for x:
7x - 8x = -8 - 4
-x = -12
x = 12
Now that we have determined x, we can find the value of y using the given information:
OL = 3y - 6
Since opposite sides in a parallelogram are congruent, OL must also be equal to NM:
3y - 6 = x - 5
Substitute x = 12 into the equation:
3y - 6 = 12 - 5
3y - 6 = 7
3y = 7 + 6
3y = 13
y = 13/3
Therefore, LMNO must be a parallelogram when x = 12 and y = 13/3.
To determine the values of x and y for which LMNO must be a parallelogram, we need to use the properties of a parallelogram.
1. In a parallelogram, opposite sides are equal in length. So, we can set up the following equation:
ON = LM
8x - 8 = 7x + 4 (Substitute the given expressions for ON and LM)
2. Solve the equation for x:
8x - 7x = 4 + 8
x = 12
3. Substitute the value of x back into the expressions for ON and LM:
ON = 8(12) - 8 = 88
LM = 7(12) + 4 = 88
4. Now, we need to consider the other pair of opposite sides. In a parallelogram, opposite sides are also parallel, which means their slopes are equal.
5. The slope formula is given by: slope = (y2 - y1) / (x2 - x1)
6. Let's find the slopes of the line segment OL and line segment NM:
OL: slope = (y - (-6)) / (3 - 0) = (y + 6) / 3
NM: slope = (y - (x - 5)) / (0 - 1) = (y - x + 5) / (-1)
7. Since OL and NM are parallel, their slopes must be equal:
(y + 6) / 3 = (y - x + 5) / (-1)
8. Solve the equation for y:
-1(y + 6) = 3(y - x + 5)
-y - 6 = 3y - 3x + 15
-4y = 3x - 21
y = (3x - 21) / -4
Therefore, the values of x and y for which LMNO must be a parallelogram are:
x = 12
y = (3(12) - 21) / -4 = -9
To determine the values of x and y for which quadrilateral LMNO must be a parallelogram, we need to use properties of parallelograms.
One property of parallelograms is that opposite sides are congruent. Therefore, we can set up the following equation for the given sides:
LM = NO (opposite sides are congruent)
Given:
LM = 7x + 4
NO = ?
To find NO, we need to use the information given about ON. We know that ON = 8x - 8.
Since NO is the opposite side of LM, it should have the same length. Therefore, we can set up the following equation:
7x + 4 = 8x - 8
Now, let's solve for x.
7x + 4 = 8x - 8 (subtract 7x from both sides)
4 = x - 8 (add 8 to both sides)
12 = x
We have found the value of x, which is x = 12.
Now that we know x, we can find the value of NO.
NO = 8x - 8
NO = 8(12) - 8
NO = 96 - 8
NO = 88
So, the length of NO is 88.
Now, let's move on to finding the value of y.
Another property of parallelograms is that opposite angles are congruent. Therefore, we can set up the following equation for the given angles:
LOM = LNO (opposite angles are congruent)
Given:
LOM = 3y - 6
LNO = ?
To find LNO, we can use the fact that opposite angles are congruent. Therefore:
LOM = LNO (opposite angles are congruent)
Now, let's set up the equation with the given values:
3y - 6 = 7x + 4 (substituting LOM with its value)
Since we know that x = 12, we can substitute it into the equation:
3y - 6 = 7(12) + 4
Now, let's solve for y.
3y - 6 = 84 + 4 (perform the multiplication)
3y - 6 = 88 (add 6 to both sides)
3y = 94 (divide both sides by 3)
y = 31.33 (approximately)
Therefore, the values of x and y for which LMNO must be a parallelogram are x = 12 and y = 31.33 (approximately).