If area of rhombus=120 and perimeter=72. Find ratio of (d1+d2) and (d1-d2)

Given:
- The area of the rhombus \( A = 120 \)
- The perimeter of the rhombus \( P = 72 \)

We are asked to find the ratio of \( (d_1 + d_2) \) to \( (d_1 - d_2) \), where \( d_1 \) and \( d_2 \) are the diagonals of the rhombus.

### Step 1: Find the side length of the rhombus

The perimeter \( P \) of a rhombus is given by:

\[
P = 4 \times s
\]

where \( s \) is the length of each side. Using the given perimeter:

\[
72 = 4 \times s \implies s = \frac{72}{4} = 18
\]

### Step 2: Use the area formula to relate diagonals

The area \( A \) of a rhombus can also be expressed in terms of its diagonals \( d_1 \) and \( d_2 \) as:

\[
A = \frac{1}{2} \times d_1 \times d_2
\]

Given that \( A = 120 \), we have:

\[
120 = \frac{1}{2} \times d_1 \times d_2 \implies d_1 \times d_2 = 240
\]

### Step 3: Use the Pythagorean theorem for diagonals and side length

For a rhombus, the diagonals bisect each other at right angles, forming four right triangles. The relationship between the diagonals and the side length \( s \) is:

\[
s^2 = \left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2
\]

Substitute \( s = 18 \) and simplify:

\[
18^2 = \left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2 \implies 324 = \frac{d_1^2}{4} + \frac{d_2^2}{4}
\]

Multiplying both sides by 4 to eliminate the denominators:

\[
1296 = d_1^2 + d_2^2
\]

### Step 4: Solve the system of equations

We now have two equations:

1. \( d_1 \times d_2 = 240 \)
2. \( d_1^2 + d_2^2 = 1296 \)

Let’s solve this system to find \( d_1 \) and \( d_2 \).

#### Step 4.1: Expressing \( d_1^2 + d_2^2 \) in terms of sum and product of \( d_1 \) and \( d_2 \)

We can use the identities:

\[
d_1^2 + d_2^2 = (d_1 + d_2)^2 - 2d_1d_2
\]

Substitute \( d_1d_2 = 240 \):

\[
1296 = (d_1 + d_2)^2 - 2 \times 240
\]

\[
1296 = (d_1 + d_2)^2 - 480
\]

\[
1776 = (d_1 + d_2)^2
\]

\[
d_1 + d_2 = \sqrt{1776} \approx 42.14
\]

#### Step 4.2: Solving for \( d_1 - d_2 \)

We can also use the identity:

\[
d_1^2 + d_2^2 = (d_1 - d_2)^2 + 2d_1d_2
\]

Substitute \( d_1d_2 = 240 \) and \( d_1^2 + d_2^2 = 1296 \):

\[
1296 = (d_1 - d_2)^2 + 480
\]

\[
816 = (d_1 - d_2)^2
\]

\[
d_1 - d_2 = \sqrt{816} \approx 28.57
\]

### Step 5: Find the ratio

Now we can compute the ratio of \( (d_1 + d_2) \) to \( (d_1 - d_2) \):

\[
\text{Ratio} = \frac{d_1 + d_2}{d_1 - d_2} = \frac{42.14}{28.57} \approx 1.475
\]

### Final Answer:
The ratio of \( (d_1 + d_2) \) to \( (d_1 - d_2) \) is approximately \( 1.475 \).

Would you like further details on any part of the solution?

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Here are some related questions you might find useful:
1. How do you derive the formula for the area of a rhombus using diagonals?
2. What are the properties of a rhombus' diagonals?
3. How does the Pythagorean theorem apply to a rhombus?
4. How do you solve quadratic systems involving sums and products of variables?
5. How do you calculate the perimeter of a rhombus given its side length?

**Tip**: Always remember that the diagonals of a rhombus bisect each other at right angles, making each half of the diagonals the legs of a right triangle.

If area of rhombus = 120 and perimeter = 72, find the ratio of (d1 + d2) and (d1 - d2).

This step-by-step guide shows how to find the ratio of (d1 + d2) to (d1 - d2) for a rhombus with an area of 120 and a perimeter of 72. Learn about rhombus properties, Pythagorean theorem, and key algebraic identities. Suitable for Grades 9-11 students.

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