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.
In a parallelogram, opposite sides are equal in length. Since DEFG is a parallelogram, we can set up the following equations:
DH = GH
x + 1 = 3x - 4
Solving this equation, we get:
2x = 5
x = 5/2
HF = HE
3y = 5y + 1
Solving this equation, we get:
-2y = 1
y = -1/2
Therefore, the values of x and y for which DEFG must be a parallelogram are x = 5/2 and y = -1/2.
Let's denote the number of sides of the regular polygon as n.
In a regular polygon, the sum of all exterior angles is always 360°.
Since each exterior angle measures 72°, we can set up the following equation:
72n = 360
To solve for n, we divide both sides of the equation by 72:
n = 360/72
n = 5
Therefore, a regular polygon with each exterior angle measuring 72° has 5 sides.
The sum of the measures of all exterior angles of any polygon, including a triangle, is always 360°.
Let's denote the measures of the two given exterior angles as x and y.
We know that x + y = 264°, which represents the sum of these two angles.
To find the measure of the third exterior angle, we can subtract the sum of the measures of the two given exterior angles from 360°:
360° - (x + y) = third exterior angle
Simplifying this equation, we get:
360° - 264° = third exterior angle
96° = third exterior angle
Therefore, the measure of the third exterior angle is 96°.
To find the number of sides, denoted as s, we can use the formula for the sum of the interior angles of a polygon:
Sum of interior angles = (s-2) * 180°
In this case, we are given that the sum of the interior angles is 2,340°.
Therefore, we can set up the following equation:
(s-2)*180° = 2,340°
Dividing both sides by 180°, we get:
s - 2 = 13
Adding 2 to both sides, we have:
s = 15
Therefore, the polygon has 15 sides.
In a parallelogram, opposite sides are equal in length. Since LMNO is a parallelogram, we can set up the following equation:
ON = LM
8x - 8 = 7x + 4
x = 12
We also know that NO = LM
NM = LM
x - 5 = 7x + 4
6x = -9
x = -3/2
Therefore, x = 12 and x = -3/2, which means LMNO must be a parallelogram.
Now, let's find the value of y.
OL = NM
3y - 6 = x - 5
3y - 6 = (-3/2) - 5
3y = -17/2
y = -17/6
Therefore, the values of x and y for which LMNO must be a parallelogram are x = 12 and y = -17/6.