Question

In parallelogram DEFG, DH = x + 1, HF = 3y, G H = 3 x − 4 , and HE = 5y + 1. Find the values of x and y. The diagram is not drawn to scale.

Answer 1

Upon solving the equations derived from the properties of a parallelogram, we find that the values of x and y are 2.5 and -0.5, respectively.

Step-by-step explanation:

The question involves solving for the variables x and y in a parallelogram DEFG. Given that DH = x + 1, HF = 3y, GH = 3x - 4, and HE = 5y + 1, we can use the properties of a parallelogram to set up equations. In parallelograms, opposite sides are equal in length. Hence, DH = GH and HF = HE.

Setting up the equations: x + 1 = 3x - 4 and 3y = 5y + 1.

Solving the equation for x: x + 1 = 3x - 4, we get 1 + 4 = 3x - x, which simplifies to 5 = 2x, therefore x = 2.5.

Solving the equation for y: 3y = 5y + 1, we get -1 = 5y - 3y, which simplifies to -1 = 2y, therefore y = -0.5.

Therefore, the values of x and y which satisfy the conditions of parallelogram DEFG are x = 2.5 and y = -0.5.

Answer 2

see explanation

Step-by-step explanation:

Using the property of parallelograms

• The diagonals bisect each other, hence

DH = HF and GH = HE

x + 1 = 3y and 3x - 4 = 5y + 1 ⇒ 3x = 5y + 5

Solving the 2 equations simultaneously

x + 1 = 3y → (1)

3x = 5y + 5 → (2)

rearrange (1) expressing x in terms of y

x = 3y - 1 → (3)

substitute x = 3y - 1 in (2)

3(3y - 1) = 5y + 5

9y - 3 = 5y + 5 ( subtract 5y from both sides )

4y - 3 = 5 ( add 3 to both sides )

4y = 8 ( divide both sides by 4 )

y = 2

substitute y = 2 into (3)

x = (3 × 2) - 1 = 6 - 1 = 5

Hence x = 5, y = 2