Answer:
Since the diagonals of rectangle MATH bisect each other and are congruent, we can use this information to find the values of x and y.
Let's label the midpoint of diagonal MT as point P. Since the diagonals bisect each other, we know that EP = PH and EM = MH.
We also know that the length of each diagonal is 36 inches, so we can use the Pythagorean theorem to find the length of segment ME:
ME^2 = EM^2 + EP^2
ME^2 = (2x+4y)^2 + (4x-y)^2
ME^2 = 4x^2 + 16xy + 16y^2 + 16x^2 - 8xy + y^2
ME^2 = 20x^2 + 8y^2 + 8xy
ME^2 = (18)^2
Simplifying this equation, we get:
20x^2 + 8y^2 + 8xy - 324 = 0
We can simplify this equation further by dividing both sides by 4:
5x^2 + 2y^2 + 2xy - 81 = 0
Now, we can use simultaneous equations to solve for x and y. We know that EM = MH, so we can set 2x+4y equal to 4x-y:
2x+4y = 4x-y
3y = 2x
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Answers & Comments
Answer:
Since the diagonals of rectangle MATH bisect each other and are congruent, we can use this information to find the values of x and y.
Let's label the midpoint of diagonal MT as point P. Since the diagonals bisect each other, we know that EP = PH and EM = MH.
We also know that the length of each diagonal is 36 inches, so we can use the Pythagorean theorem to find the length of segment ME:
ME^2 = EM^2 + EP^2
ME^2 = (2x+4y)^2 + (4x-y)^2
ME^2 = 4x^2 + 16xy + 16y^2 + 16x^2 - 8xy + y^2
ME^2 = 20x^2 + 8y^2 + 8xy
ME^2 = (18)^2
Simplifying this equation, we get:
20x^2 + 8y^2 + 8xy - 324 = 0
We can simplify this equation further by dividing both sides by 4:
5x^2 + 2y^2 + 2xy - 81 = 0
Now, we can use simultaneous equations to solve for x and y. We know that EM = MH, so we can set 2x+4y equal to 4x-y:
2x+4y = 4x-y
3y = 2x