HOW TO FIND THE LENGTH OF DIAGONAL OF A PARALLELOGRAM

Problem 1 :

For the parallelogram given below, solve for m and n.

Solution :

Since the given shape is a parallelogram, the diagonals will bisect each other.

Problem 2 :

For the parallelogram given below, solve for j and k.

Solution :

Problem 3 :

In parallelogram PQRS, solve for x and y.

Solution :

Problem 4 :

Solve the n in the following parallelogram ABCD. Find length of the diagonal BD.

Solution :

Diagonals will bisect each other.

3n - 6 = 5n - 122

3n - 5n = -122 + 6

-2n = -116

Dividing by 2 on both sides.

n = 116/2

n = 58

Length of BD = 3n - 6 + 5n - 122

= 8n - 128

= 8(58) - 128

= 464 - 128

= 336

So, length of the diagonal BD is 336.

Problem 5 :

Length of the longer diagonal in the parallelogram FAST.

Solution :

FO = 5.5, then FS = 5.5(2) ==>  11

TO = 3x - 1, OA = 2x + 7

3x - 1 = 2x + 7

3x - 2x = 7 + 1

x = 8

Length of diagonal TA = 3x - 1 + 2x + 7

= 5x + 6

= 5(8) + 6

= 40 + 6

= 46

So, length of the longer diagonal is 46.

Problem 6 :

For what value of x is quadrilateral MNPQ a parallelogram? Explain your reasoning

Solution :

In parallelogram, the diagonals will bisect each other. From the parallelogram above,

10 - 3x = 2x

10 = 2x + 3x

10 = 5x

x = 10/5

x = 2

So, the value of x is 2.

Problem 7 :

find the value of x that makes the quadrilateral a parallelogram.

Solution :

4x + 2 = 5x - 6

4x - 5x = -6 - 2

-1x = -8

x = 8

So, the value of x is 8.

Problem 8 :

Solution :

6x = 3x + 2

6x - 3x = 2

3x = 2

x = 2/3

So, the value of x is 2/3.

Problem 9 :

What value of x makes the quadrilateral a parallelogram? Explain how you found your answer.

Solution :

In parallelogram opposite sides will be equal, diagonals will bisect each other.

Equating the opposite sides :

Equating the diagonals :

In all of the ways, we receive the same answer.

Problem 10 :

Find the value of each variable in the parallelogram.

Solution :

Problem 11 :

Find the coordinates of the intersection of the diagonals of ▱QRST with vertices Q(−8, 1), R(2, 1), S(4, −3), and T(−6, −3).

Solution :

In the parallelogram QRST, QS and TR are diagonals.

Since the diagonals bisect each other, the point where both diagonals are intersecting each other is the mid point of the diagonals.

Midpoint = (x₁ + x₂)/2, (y₁ + y₂)/2

Midpoint of QS = (-8 + 4)/2, (1 - 3)/2

= (-4/2, -2/2)

= (-2, -1)

Midpoint of TR = (2 - 6)/2, (1 - 3)/2

= (-4/2, -2/2)

= (-2, -1)

Problem 12 :

As shown in the diagram below, the diagonals of parallelogram QRST intersect at E. If QE = x² + 6x, SE = x + 14, and TE = 6x − 1, determine TE algebraically

Solution :

x² + 6x = x + 14

x² + 6x - x - 14 = 0

x² + 5x - 14 = 0

x² + 7x - 2x - 14 = 0

x(x + 7) - 2(x + 7) = 0

(x - 2)(x + 7) = 0

x - 2 = 0  and x + 7 = 0

x = 2 and x = -7

So, the value of x is 2.

TE = 6(2) - 1

= 12 - 1

= 11

So, the length TE is 11.