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The figure is parallelogram D E F G with diagonals that intersect at point H
In parallelogram DEFG, DH = x + 1, HF = 3y, GH = 3x − 4,
and HE = 5y + 1. Find the values of x and y. The diagram is not drawn to scale.
Since DEFG is a parallelogram, its diagonals DE and FG bisect each other. Therefore, DH = HG and HF = HE.
From the given information, we have:
DH = x + 1 (1)
HF = 3y (2)
GH = 3x - 4 (3)
HE = 5y + 1 (4)
Since DH = HG, we can equate equations (1) and (3):
x + 1 = 3x - 4
4 = 2x
x = 2
Replacing x = 2 in equation (3):
GH = 3(2) - 4
GH = 6 - 4
GH = 2
Since HF = HE, we can equate equations (2) and (4):
3y = 5y + 1
-2y = 1
y = -1/2
Therefore, the values of x and y are x = 2 and y = -1/2.
To find the values of x and y, we will use the properties of parallelograms and the information given about the lengths of the diagonals.
1. Start by writing down the given information:
DH = x + 1
HF = 3y
GH = 3x - 4
HE = 5y + 1
2. Since diagonals of a parallelogram bisect each other, we know that DH + GH = HF + HE. Substitute the given values to get the equation:
(x + 1) + (3x - 4) = 3y + (5y + 1)
3. Simplify the equation:
4x - 3 = 8y + 1
4. Rearrange the equation to isolate one variable on one side:
4x - 8y = 4
5. Now, we will use another property of parallelograms which states that the opposite sides are congruent. Therefore, DH = GF and GH = EF. We can set up two more equations by substituting the given values:
x + 1 = 3x - 4 (equation 1)
3x - 4 = 5y + 1 (equation 2)
6. Solve equation 1 for x:
subtract x from both sides of the equation to get:
1 = 2x - 4
add 4 to both sides of the equation to get:
5 = 2x
divide both sides by 2 to solve for x:
x = 5/2
7. Substitute the value of x = 5/2 into equation 2:
3(5/2) - 4 = 5y + 1
8. Simplify the equation:
15/2 - 4 = 5y + 1
9. Simplify further:
15/2 - 8/2 = 5y + 1
10. Combine like terms on the left side:
7/2 = 5y + 1
11. Subtract 1 from both sides of the equation:
7/2 - 1 = 5y
12. Simplify the equation:
5/2 = 5y
13. Divide both sides by 5 to solve for y:
y = 5/10
14. Simplify the fraction:
y = 1/2
Finally, we have found the values of x and y:
x = 5/2 and y = 1/2
To find the values of x and y in the parallelogram DEFG, we can use the properties of parallelograms and the given information.
First, we know that in a parallelogram, opposite sides are congruent.
Therefore, DG = HF, and DH = GF.
From the given information, we can set up two equations:
Equation 1: DG = HF
Equation 2: DH = GF
Substituting the given values into the equations:
Equation 1: 3x - 4 = 3y
Equation 2: x + 1 = 5y + 1
Now we have a system of two equations with two variables (x and y). We can solve this system of equations to find the values of x and y.
Let's solve the equations step by step:
Step 1: Solve Equation 2 for x:
x = 5y + 1 - 1
x = 5y
Step 2: Substitute the value of x from Equation 2 into Equation 1:
3(5y) - 4 = 3y
Step 3: Simplify and solve for y:
15y - 4 = 3y
15y - 3y = 4
12y = 4
y = 4/12
y = 1/3
Step 4: Substitute the value of y back into Equation 2 to find x:
x = 5(1/3)
x = 5/3
Therefore, the values of x and y are x = 5/3 and y = 1/3 respectively.