Question

The perimeter of a rhombus is 52 cm and one of its diagonals is 10 cm. Find the length of the other diagonal and the area of the rhombus.

Answer 1

Given that perimeter of the rhombus = 52cm.

We know that perimeter of the rhombus = 4a

        4a = 52

          a = 52/4

             = 13.


Therefore Each side of the rhombus = 13cm.

Length of the one diagonal = 10cm.

Then the length of the other diagonal will be:

= $2 * \sqrt{13^2 - ( \frac{10}{2})^2 }$

= $2 * \sqrt{169 - 5^2}$

= $2 * \sqrt{169 - 25}$

= $= 2 * \sqrt{144}$

$= 2 * 12$

= 24cm.


We know that Area of the rhombus = 1/2 * product of the diagonals

                                                            = 1/2 * 10 * 24

                                                            = 24 * 5

                                                            = 120cm^2.


Hope this helps!

Answer 2

Length of other diagonal is 24 cm and area of rhombus is 120 cm².

Step by step Explanation:

Given:

• The perimeter of a rhombus is 52 cm and
• One of its diagonals is 10 cm.

To find:

• Find the length of the other diagonal and
• The area of the rhombus.

Solution:

Formula/Concept to be used:

1. Perimeter of rhombus$\bf = 4a$; where 'a' is side of rhombus.
2. Area of rhombus$\bf = \frac{1}{2} d_1d_2$; where d1 and d2 are the diagonals of rhombus.
3. Diagonals of rhombus bisects each other at right angle .
4. Pythagoras theorem: In right triangle Hypotenuse² = Base² +Perpendicular²

Step 1:

Calculate side of rhombus.

We know that

Perimeter of rhombus= 4a

$4a = 52$

or

$a = \frac{52}{4}$

or

$\bf a = 13 \: cm$

Step 2:

Calculate diagonal of rhombus.

Draw the figure.

*see the attachment.

Apply Pythagoras theorem in right triangle AOB.

${AB}^{2} = {OA}^{2} + {OB}^{2}$

or

$( {13)}^{2} = {(5)}^{2} + {x}^{2}$

or

${x}^{2} = 169 - 25$

or

${x}^{2} = 144$

or

$\bf x = 12 \: cm$

Now,

$\bf BD=DO+OB$

or

$BD=12 + 12$

or

$\bf BD=24 \: cm$

Thus,

Length of other diagonal is 24 cm.

Step 3:

Calculate area of rhombus.

Area of rhombus$= \frac{1}{2} \times 10 \times 24$

or

Area of rhombus$\bf = 120 \: {cm}^{2}$

Thus,

Length of other diagonal is 24 cm and area of rhombus is 120 cm².

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