Question

44. The area of a rhombus is 480 cm", and one of its diagonals measures
48 cm. Find (i) the length of the other diagonal, (ii) the length of each of
its sides, and (iii) its perimeter.
of wl
ANSWERS EYERCISE 14​

Answer 1

Answer:

20 cm

26 cm

104 cm

Step-by-step explanation:

the formula of area by diagonal is 1/2 × diagonal 1 × diagonal 2

so by applying formula

480 = 1/2 × 48 × X

X = 480×2/48

X = 20 cm

Answer 2

━━━━━━━━━━━━━━━━━━━━

✦ Required Answer:

♦️ GiveN:

• Area of the rhombus = 480 cm^2
• One of the diagonal = 48 cm

♦️ To FinD:

• Length of other diagonal
• Length of each side.
• Perimeter of the rhombus

━━━━━━━━━━━━━━━━━━━━

✦ Explanation of Concepts:

The question is based on multiple concepts and formulas but they are very simple and easy to understand. Here are some formulae to be used to solve this question.

$\large{ \dag{ \rm{ \red{Area \: of \: rhombus \: when \: diagonals \: given}}}} \\ \large{ \boxed{ \rm{= \frac{1}{2} \times diagonal_1 \times diagonal_2}}}$

Next is how to find the sides when diagonal are given,

The diagonal of a rhombus perpendicularly bisect each other, So they form a right angled triangle with half of the diagonals as their base and perpendicular and the hypotenuse is the side of the rhombus.

⏺ Refer to the attachment....

So, the side of the rhombus(by using pythagoras theoram)

$\large{ \rm{ \dashrightarrow{ {side}^{2} = {( \frac{d_1}{2}) }^{2} + {( \frac{d_2}{2}) }^{2}}} } \\ \\ \large{ \rm{ \dashrightarrow \: side = \sqrt{ \frac{ {d_1}^{2} }{4} + \frac{ {d_2}^{2} }{4}}}} \\ \\ \large{ \rm{ \dashrightarrow \: \boxed{ \rm{side = \frac{1}{2} \sqrt{d_1 {}^{2} + d_2 {}^{2} } }}}}$

By using these formula, we can solve this question.

━━━━━━━━━━━━━━━━━━━━

✦ Solution:

$\large{\red{\rm{(1)}}}$ We know, area of rhombus = 1/2 × diagonal 1 × diagonal 2.

So, we can write 1/2× diagonal 1 × diagonal 2 = 480 cm^2

Now, Putting the given value of diagonal 1,

$\large{ \rm{ \dashrightarrow \: \frac{1}{2} \times 48 \times d_2 = 480 {cm}^{2} }} \\ \\ \large{ \rm{ \dashrightarrow \: d_2 = \cancel{\frac{480 \times 2}{48} \: cm}}} \\ \\ \large{ \rm{ \dashrightarrow \: d_2 = \boxed{ \rm{ \green{20 \: cm}}}}}$

The length of other diagonal = 20 cm

━━━━━━━━━━━━━━━━━━━━

$\large{\rm{\red{(2)}}}$ We have already derived the formula for finding side when diagonals are given.

By using formula,

$\large{ \rm{ \dashrightarrow \: Side = \frac{1}{2} \times \sqrt{d_1 {}^{2} + d_2 {}^{2} }}} \\ \\ \large{ \rm{ \dashrightarrow \: Side = \frac{1}{2} \times \sqrt{ {48}^{2} + {20}^{2} } \: cm }} \\ \\ \large{ \rm{ \dashrightarrow \: Side = \frac{1}{2} \times \sqrt{ {52}^{2} } \: cm }} \\ \\ \large{ \rm{ \dashrightarrow \: Side = \frac{1}{2} \times 52 \: cm}} \\ \\ \large{ \rm{ \dashrightarrow \: Side = \boxed{ \rm{ \green{26 \: cm}}}}}$

So, Length of the side = 26 cm

━━━━━━━━━━━━━━━━━━━━

$\large{\rm{\red{(3)}}}$ All the sides of the rhombus are equal, hence perimeter = 4 × sides.

We already have the length of sides,

By applying formula,

$\large{ \rm{ \dashrightarrow \: Perimeter = 4 \times side}} \\ \\ \large{ \rm{ \dashrightarrow \: Perimeter = 4 \times 26 \: cm}} \\ \\ \large{ \rm{ \dashrightarrow \: Perimeter = \boxed{ \rm{ \green{104 \: cm}}}}}$

So, Perimeter of the rhombus = 104 cm.

$\large{ \therefore{ \underline{ \underline{ \pink{ \rm{Hence,\: solved \: \dag}}}}}}$

━━━━━━━━━━━━━━━━━━━━