Answer:
A: x = 9 4/7
B: y = 15
C: P = 95 units
Explanation:
You want the values of x and y, and the perimeter of the isosceles trapezoid ABCD with AB=20, CD=2y-10, AD=15, BC=40, AD║CB, angle A=10x-22, and angle C=4x+68.
Part A
The value of x can be found using the fact that base angles are congruent and the sum of angles on one side of the trapezoid is 180°.
∠A +∠C = 180
(10x -22) +(4x +68) = 180 . . . . . use the given angle expressions
14x +46 = 180 . . . . . . . . . . . . simplify
14x = 134 . . . . . . . . . . . . . . subtract 46
x = 134/14 = 9 4/7 . . . . divide by 14
The value of x is 9 4/7.
Part B
Sides of the isosceles trapezoid are congruent
CD = AB
2y -10 = 20 . . . . . substitute the given lengths
2y = 30 . . . . . . add 10
y = 15 . . . . . divide by 2
The value of y is 15.
Part C
The perimeter is the sum of the side lengths.
P = AB +BC +CD +AD
P = 20 +40 +20 +15 = 95
The perimeter is 95 units.
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Additional comment
The expressions for the angles and the value of x are impossible in this figure. For the found value of x, angle A is 73 5/7°, and angle C is 106 2/7°. The angle adjacent to the short base will be obtuse, so these are impossible measures.
For the given side lengths, angle A is 128.7°, and angle C is 51.3° (using trigonometry).