Let D1 and d2 be the diagonal of rhombus with an area of 120 cm2 and perimeter 72 cm find the ratio of D1+ d2 to that of d1-d2

Let the diagonals of the rhombus be \( D_1 \) and \( D_2 \), and we are given:

- The area of the rhombus is \( 120 \, \text{cm}^2 \).
- The perimeter of the rhombus is \( 72 \, \text{cm} \).

### Step 1: Relationship Between Diagonals and Area
The area \( A \) of a rhombus is given by the formula:
\[
A = \frac{1}{2} \times D_1 \times D_2
\]
Substituting the given area:
\[
120 = \frac{1}{2} \times D_1 \times D_2
\]
Multiplying both sides by 2:
\[
240 = D_1 \times D_2
\]
Thus, the product of the diagonals is:
\[
D_1 \times D_2 = 240
\]

### Step 2: Relationship Between Diagonals and Perimeter
The perimeter \( P \) of a rhombus is given by:
\[
P = 4s
\]
where \( s \) is the side length of the rhombus. The side length \( s \) is related to the diagonals by:
\[
s = \sqrt{\left( \frac{D_1}{2} \right)^2 + \left( \frac{D_2}{2} \right)^2}
\]
We are given \( P = 72 \), so:
\[
4s = 72 \implies s = 18
\]
Now, substituting \( s = 18 \) into the equation for \( s \):
\[
18 = \sqrt{\left( \frac{D_1}{2} \right)^2 + \left( \frac{D_2}{2} \right)^2}
\]
Squaring both sides:
\[
324 = \left( \frac{D_1}{2} \right)^2 + \left( \frac{D_2}{2} \right)^2
\]
Multiplying by 4:
\[
1296 = D_1^2 + D_2^2
\]

### Step 3: Solving for the Ratio \( \frac{D_1 + D_2}{D_1 - D_2} \)

We now have the following two equations:
\[
D_1 \times D_2 = 240
\]
\[
D_1^2 + D_2^2 = 1296
\]
Let \( x = D_1 + D_2 \) and \( y = D_1 - D_2 \). We know the following identities:
\[
D_1^2 + D_2^2 = \frac{x^2 + y^2}{2}
\]
\[
D_1 \times D_2 = \frac{x^2 - y^2}{4}
\]
We can solve this system to find the ratio. 

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Let D1 and D2 be the diagonals of a rhombus with an area of 120 cm² and a perimeter of 72 cm. Find the ratio of D1 + D2 to D1 - D2.

Learn how to find the ratio of the sum and difference of the diagonals of a rhombus with a given area of 120 cm² and perimeter of 72 cm. This step-by-step solution covers geometric concepts, algebraic manipulation, and quadratic identities. Suitable for students in Grades 9-11.

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