The values of x and y that satisfy the given conditions are x=12and y=19.67 (rounded to two decimal places).
Step-by-step explanation:
In an isosceles trapezoid, the base angles are congruent. Therefore, we have:
mL = mE = 68
mLA = mEH = 6y-8 (opposite angles are congruent)
mLB = mHA (base angles are congruent)
m2H = 5x + 12
Since the sum of angles in a trapezoid is 360 degrees, we can use this information to set up an equation:
mL + mLA + mEH + m2H = 360
Substituting in the values we have:
68 + (6y-8) + (6y-8) + (5x+12) = 360
Combine like terms:
5x + 12 + 12y - 16 = 292
Simplify:
5x + 12y - 4 = 292
Combine like terms:
5x + 12y = 296
Since we have two variables and one equation, we need more information to solve for x and y. However, we can use the fact that the sum of the lengths of the two non-parallel sides of an isosceles trapezoid is equal to the sum of the lengths of the parallel sides. In other words, we have:
LE + AH = EA + LH
Substituting in the values we have:
LE + AH = 68 + (5x+12)
EA + LH = 68 + 68
Simplifying:
LE + AH = 5x + 80
EA + LH = 136
Since LE = AH (opposite sides of an isosceles trapezoid are congruent), we can substitute and simplify:
2LE = 5x + 80 - 2AH
2LE = 5x + 80 - 2LE
4LE = 5x + 80
LE = (5x + 80)/4
Since we know that LE is a positive integer length, we can set up an equation to solve for x:
LE = (5x + 80)/4 = k, where k is some positive integer
Multiplying both sides by 4:
5x + 80 = 4k
Subtracting 80 from both sides:
5x = 4k - 80
Dividing both sides by 5:
x = (4/5)k - 16
Since x is a positive integer, we can choose k to be a multiple of 5 that is greater than or equal to 80/4 + 16 = 36. The smallest such value is k = 40, which gives us:
x = (4/5)k - 16 = (4/5)(40) - 16 = 12
Now we can use the equation we derived earlier:
5x + 12y = 296
Substituting x = 12:
5(12) + 12y = 296
Simplifying:
60 + 12y = 296
Subtracting 60 from both sides:
12y = 236
Dividing both sides by 12:
y = 19.67
Therefore, the values of x and y that satisfy the given conditions are x = 12 and y = 19.67 (rounded to two decimal places).
Answers & Comments
Answer:
The values of x and y that satisfy the given conditions are x = 12 and y = 19.67 (rounded to two decimal places).
Step-by-step explanation:
In an isosceles trapezoid, the base angles are congruent. Therefore, we have:
mL = mE = 68
mLA = mEH = 6y-8 (opposite angles are congruent)
mLB = mHA (base angles are congruent)
m2H = 5x + 12
Since the sum of angles in a trapezoid is 360 degrees, we can use this information to set up an equation:
mL + mLA + mEH + m2H = 360
Substituting in the values we have:
68 + (6y-8) + (6y-8) + (5x+12) = 360
Combine like terms:
5x + 12 + 12y - 16 = 292
Simplify:
5x + 12y - 4 = 292
Combine like terms:
5x + 12y = 296
Since we have two variables and one equation, we need more information to solve for x and y. However, we can use the fact that the sum of the lengths of the two non-parallel sides of an isosceles trapezoid is equal to the sum of the lengths of the parallel sides. In other words, we have:
LE + AH = EA + LH
Substituting in the values we have:
LE + AH = 68 + (5x+12)
EA + LH = 68 + 68
Simplifying:
LE + AH = 5x + 80
EA + LH = 136
Since LE = AH (opposite sides of an isosceles trapezoid are congruent), we can substitute and simplify:
2LE = 5x + 80 - 2AH
2LE = 5x + 80 - 2LE
4LE = 5x + 80
LE = (5x + 80)/4
Since we know that LE is a positive integer length, we can set up an equation to solve for x:
LE = (5x + 80)/4 = k, where k is some positive integer
Multiplying both sides by 4:
5x + 80 = 4k
Subtracting 80 from both sides:
5x = 4k - 80
Dividing both sides by 5:
x = (4/5)k - 16
Since x is a positive integer, we can choose k to be a multiple of 5 that is greater than or equal to 80/4 + 16 = 36. The smallest such value is k = 40, which gives us:
x = (4/5)k - 16 = (4/5)(40) - 16 = 12
Now we can use the equation we derived earlier:
5x + 12y = 296
Substituting x = 12:
5(12) + 12y = 296
Simplifying:
60 + 12y = 296
Subtracting 60 from both sides:
12y = 236
Dividing both sides by 12:
y = 19.67
Therefore, the values of x and y that satisfy the given conditions are x = 12 and y = 19.67 (rounded to two decimal places).