How To Find The Diagonal Of A Parallelogram Without Angles

Diagonal of Parallelogram

The diagonal of a parallelogram is the line segment that connects its not-adjacent vertices. A parallelogram has ii diagonals and the length of the diagonals of a parallelogram tin exist found past using various formulas depending on the given parameters and dimensions. Permit us larn more about the diagonals of a parallelogram in this commodity.

i.	What is the Diagonal of a Parallelogram?
2.	Diagonal of Parallelogram Formula
3.	FAQs on Diagonal of Parallelogram

What is the Diagonal of Parallelogram?

The diagonals of a parallelogram tin be drawn past joining the two non-next vertices of the parallelogram. It should be noted that the 2 diagonals of a parallelogram bifurcate each other and they carve up the parallelogram into congruent triangles.

Diagonal of Parallelogram Formula

The formula for the diagonals of a parallelogram is used to summate the length of the diagonals of a given parallelogram. There are different formulas for unlike kinds of parallelograms. Observe the effigy given below which shows a parallelogram along with its diagonals. Hither 'p' and 'q' are the diagonals and 'ten' and 'y' are the ii sides of the parallelogram.

diagonals of parallelogram formula

The unproblematic formula for finding the length of the diagonals of a parallelogram is given below. For this formula, we demand the length of the sides and whatsoever of the known angles. If nosotros follow the figure given above, nosotros tin can notice that:

p and q are taken to be the length of the diagonals respectively.
x and y are the sides of the parallelogram.
Angle A and Bending B are two interior angles of the parallelogram.

Formula 1: For any parallelogram, the formula for the length of the diagonals is expressed as:

\(p = \sqrt{ten^two + y^2 - 2xy \cos A} = \sqrt{x^2 + y^ii + 2xy \cos B} \)
\(q = \sqrt{x^2 + y^2 + 2xy \cos A} = \sqrt{x^ii + y^2 - 2xy \cos B} \)

Formula two: Another formula which expresses the relationship between the length of the diagonals and sides of the parallelogram is:

p^two + q^two= 2(ten²+ y²)

Where,

p and q are the diagonals respectively.
x and y are the sides of the parallelogram.

It should be noted that a square, a rectangle, and a rhomb come nether the category of parallelograms. And since they have different properties, the formula that is used to observe their diagonals is likewise dissimilar. For case, the diagonal of a square (d) = a√2; where 'd' is the diagonal and 'a' is the side of the foursquare. The diagonal of a rectangle (d) = √( l² + w²), where l = length of the rectangle and west = width of the rectangle. Therefore, the formula for the diagonal of a parallelogram varies for different kinds of parallelograms.

Properties of Diagonal of Parallelogram

The following points prove the properties of the diagonals of a parallelogram. Since a parallelogram includes a square, a rectangle, a rhombus, the diagonals of these figures have a few common properties and a few different ones.

The diagonals of a parallelogram always bisect each other.
In a square, the diagonals are equal and bisect each other at right angles.
In a rectangle, the diagonals are equal and they bisect each other but not at correct angles.
In a rhombus, the diagonals may not be necessarily equal, but they are perpendicular to each other.

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Examples using Diagonal of Parallelogram Formula

Example 1: Notice the length of the diagonals of the rhombus of side length 4 inches, if the interior angles are 120° and lx°.

Solution:

Given, Interior bending A = 120°, and angle B = 60°.
10 = 4, y = 4

Using diagonal of parallelogram formula,

\( p = \sqrt{x^ii + y^2 - 2xy \cos A} \)
\( q = \sqrt{10^2 + y^2 + 2xy \cos A}\)
Putting the values in the formula for p:
\(\begin{align} p &= \sqrt{iv^2 + 4^2 - (two \times 4 \times four \times \cos 60)} \\ &= \sqrt{32 - xvi} \\ p &= four \end{align}\)
Now, doing same for q,
\(\begin{marshal} q &= \sqrt{4^2 + four^2 + (ii \times iv \times 4 \times \cos 60)} \\ &= \sqrt{32 + 16} \\ p &= \sqrt {48} \\ p &= 6.92 \end{align}\)

Respond: The length of the diagonals are 4 in and 6.92 in.
Example two: For a parallelogram ABCD, if the length of the adjacent sides is 35 ft and 82 ft. If one of the interior angles is 37°. Detect the length of any diagonal.

Solution:

Given, Interior bending A = 37°
x = 35 ft, y = 82 ft

Using diagonal of parallelogram formula,

\(p = \sqrt{x^2 + y^2 - 2xy \cos A}\)
Putting the values in the formula for p:
\(\begin{align} p &= \sqrt{35^2 + 82^ii - (2 \times 35 \times 82 \times \cos 37)} \\ &= \sqrt{3365} \\ p &= 58 \finish{align}\)

Reply: The length of the diagonal is 58 ft.
Case 3: Calculate the length of the diagonal of a parallelogram with sides 4 units, half-dozen units and an interior angle A which is equal to threescore degrees.

Solution:

Given, a = 4 units, b = half-dozen units, bending A = threescore°

Using diagonal of parallelogram formula,

\(p = \sqrt{x^two + y^2 - 2xy \cos A}\)
Putting the values in the formula for p:
\(p = \sqrt{4^2 + half dozen^two - 24}\)= \(\sqrt28\) = 5.291

Answer: Diagonal of parallelogram = 5.291 units.

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Practice Questions on Diagonal of Parallelogram

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FAQs on Diagonal of Parallelogram

What is the Diagonal of a Parallelogram?

The diagonal of a parallelogram is the line segment that joins the two non-side by side vertices of the parallelogram. Information technology is to exist noted that 2 diagonals tin can exist fatigued in a parallelogram.

What is the Diagonal of a Parallelogram Formula?

A uncomplicated formula which is used to find the length of the diagonals of a parallelogram needs the value of the interior angles and the length of the sides. For whatever parallelogram, the formula for the length of the diagonals is expressed every bit, \( p = \sqrt{x^2 + y^2 - 2xy \cos A} = \sqrt{ten^ii + y^2 + 2xy \cos B} \) and \( q = \sqrt{x^2 + y^two + 2xy \cos A} = \sqrt{x^ii + y^ii - 2xy \cos B} \), where p and q are the lengths of the diagonals, angle A and angle B are the given interior angles and x and y are the sides of the parallelogram.

How to Utilise the Diagonal of a Parallelogram Formula?

For whatever parallelogram, permit p and q be the lengths of the diagonals and ten and y be the sides of the parallelogram and then

Pace one: Cheque for the given parameters, the values of the sides of the parallelograms, and the corresponding angles.
Pace two: Substitute the values in the formula, \( p = \sqrt{x^ii + y^ii - 2xy \cos A} = \sqrt{x^2 + y^2 + 2xy \cos B} \) and \( q = \sqrt{10^2 + y^2 + 2xy \cos A} = \sqrt{x^2 + y^2 - 2xy \cos B} \).

What are the Components of the Diagonal of Parallelogram Formula?

The formula for the diagonal of parallelogram helps to find the length of the diagonals by using the length of the sides and whatever of the known angles. Thus, its components include the sides of the parallelogram and the corresponding angles.

Practice the Diagonals of a Parallelogram Bisect Each Other?

Yes, the diagonals of a parallelogram bifurcate each other. This means that the diagonals of a parallelogram divide each other into 2 equal parts.

Are the Diagonals of a Parallelogram Equal?

A parallelogram includes a square, a rectangle, and a rhombus. While the diagonals of a square and rectangle are equal, the diagonals of a rhomb may not exist necessarily equal.

How to Find the Diagonals of a Parallelogram without Angles?

The length of the diagonals of a parallelogram can be calculated even when the interior angles are non given. For case, if the parallelogram is a rectangle, we know that the diagonals of a rectangle form a correct-angled triangle. So, in this case, if the sides of the rectangle are known, the length of the diagonal can be calculated using the Pythagoras theorem because the diagonal becomes the hypotenuse. This method tin also be applied if the given parallelogram is a square.

Source: https://www.cuemath.com/diagonal-of-parallelogram-formula/

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