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A rhombus and a square both have sides of length [tex]$6 \, \text{cm}$[/tex]. The area of the rhombus is [tex]\frac{4}{5}[/tex] of the area of the square. Find the height of the rhombus.

Sagot :

GinnyAnswer
To determine the height of the rhombus given that both the rhombus and the square have sides of 6 cm and the area of the rhombus is \(\frac{4}{5}\) of the area of the square, follow these steps:

### Step 1: Calculate the Area of the Square
The formula for the area \(A\) of a square with side length \(s\) is:
[tex]\[ A_{\text{square}} = s^2 \][/tex]

Given the side length \(s = 6 \, \text{cm}\), we have:
[tex]\[ A_{\text{square}} = 6^2 = 36 \, \text{cm}^2 \][/tex]

### Step 2: Calculate the Area of the Rhombus
We are informed that the area of the rhombus is \(\frac{4}{5}\) of the area of the square. Using the area of the square calculated above:
[tex]\[ A_{\text{rhombus}} = \frac{4}{5} \times 36 \][/tex]

Perform the multiplication:
[tex]\[ A_{\text{rhombus}} = \frac{4 \times 36}{5} = \frac{144}{5} = 28.8 \, \text{cm}^2 \][/tex]

### Step 3: Relate the Area of the Rhombus to its Height
The area of a rhombus can also be found using the side length and the height \(h\) with the formula:
[tex]\[ A_{\text{rhombus}} = \text{side length} \times \text{height} \][/tex]

Given the side length is 6 cm and the area is 28.8 cm\(^2\):
[tex]\[ 28.8 = 6 \times h \][/tex]

### Step 4: Solve for the Height \(h\) of the Rhombus
To find \(h\), we solve the equation above:
[tex]\[ h = \frac{28.8}{6} \][/tex]

Perform the division:
[tex]\[ h = 4.8 \, \text{cm} \][/tex]

### Conclusion
The height of the rhombus is:
[tex]\[ h = 4.8 \, \text{cm} \][/tex]
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