Answer :
Let's solve the problem step-by-step.
### Step 1: Understanding the Properties of a Rhombus
A rhombus is a type of polygon that has the following characteristics:
1. All sides have the same length.
2. The diagonals bisect each other at right angles (90 degrees).
### Step 2: Given Information
- Area of the rhombus ([tex]\(A\)[/tex]) = 120 cm²
- Length of one diagonal ([tex]\(d_1\)[/tex]) = 24 cm
### Step 3: Finding the Length of the Other Diagonal ([tex]\(d_2\)[/tex])
The formula for the area of a rhombus in terms of its diagonals is:
[tex]\[ \text{Area} = \frac{1}{2} \times d_1 \times d_2 \][/tex]
We need to find [tex]\(d_2\)[/tex]. Rearranging the formula to solve for [tex]\(d_2\)[/tex]:
[tex]\[ d_2 = \frac{2 \times \text{Area}}{d_1} \][/tex]
Substitute the given values:
[tex]\[ d_2 = \frac{2 \times 120}{24} \][/tex]
[tex]\[ d_2 = \frac{240}{24} \][/tex]
[tex]\[ d_2 = 10 \, \text{cm} \][/tex]
So, the length of the other diagonal is 10 cm.
### Step 4: Finding the Length of the Side of the Rhombus
Each diagonal splits the rhombus into two right-angled triangles. In one of these triangles:
- The halves of the diagonals are the legs of the right triangle.
- The side of the rhombus is the hypotenuse.
Let's denote the side of the rhombus as [tex]\(s\)[/tex].
Using the Pythagorean theorem in one of these right triangles:
[tex]\[ s = \sqrt{\left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2} \][/tex]
Substitute the given values:
[tex]\[ s = \sqrt{\left(\frac{24}{2}\right)^2 + \left(\frac{10}{2}\right)^2} \][/tex]
[tex]\[ s = \sqrt{12^2 + 5^2} \][/tex]
[tex]\[ s = \sqrt{144 + 25} \][/tex]
[tex]\[ s = \sqrt{169} \][/tex]
[tex]\[ s = 13 \, \text{cm} \][/tex]
### Final Answer
- The length of the other diagonal ([tex]\(d_2\)[/tex]) is 10 cm.
- The length of a side of the rhombus ([tex]\(s\)[/tex]) is 13 cm.
### Step 1: Understanding the Properties of a Rhombus
A rhombus is a type of polygon that has the following characteristics:
1. All sides have the same length.
2. The diagonals bisect each other at right angles (90 degrees).
### Step 2: Given Information
- Area of the rhombus ([tex]\(A\)[/tex]) = 120 cm²
- Length of one diagonal ([tex]\(d_1\)[/tex]) = 24 cm
### Step 3: Finding the Length of the Other Diagonal ([tex]\(d_2\)[/tex])
The formula for the area of a rhombus in terms of its diagonals is:
[tex]\[ \text{Area} = \frac{1}{2} \times d_1 \times d_2 \][/tex]
We need to find [tex]\(d_2\)[/tex]. Rearranging the formula to solve for [tex]\(d_2\)[/tex]:
[tex]\[ d_2 = \frac{2 \times \text{Area}}{d_1} \][/tex]
Substitute the given values:
[tex]\[ d_2 = \frac{2 \times 120}{24} \][/tex]
[tex]\[ d_2 = \frac{240}{24} \][/tex]
[tex]\[ d_2 = 10 \, \text{cm} \][/tex]
So, the length of the other diagonal is 10 cm.
### Step 4: Finding the Length of the Side of the Rhombus
Each diagonal splits the rhombus into two right-angled triangles. In one of these triangles:
- The halves of the diagonals are the legs of the right triangle.
- The side of the rhombus is the hypotenuse.
Let's denote the side of the rhombus as [tex]\(s\)[/tex].
Using the Pythagorean theorem in one of these right triangles:
[tex]\[ s = \sqrt{\left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2} \][/tex]
Substitute the given values:
[tex]\[ s = \sqrt{\left(\frac{24}{2}\right)^2 + \left(\frac{10}{2}\right)^2} \][/tex]
[tex]\[ s = \sqrt{12^2 + 5^2} \][/tex]
[tex]\[ s = \sqrt{144 + 25} \][/tex]
[tex]\[ s = \sqrt{169} \][/tex]
[tex]\[ s = 13 \, \text{cm} \][/tex]
### Final Answer
- The length of the other diagonal ([tex]\(d_2\)[/tex]) is 10 cm.
- The length of a side of the rhombus ([tex]\(s\)[/tex]) is 13 cm.
