The height of the rhombus is 7 cm.
In-Depth Discussion of the Rhombus Height Calculation
Introduction to the Rhombus
A rhombus is a quadrilateral with all four sides of equal length. It is a special type of parallelogram, which means its opposite sides are parallel and its opposite angles are equal. Key properties of a rhombus include:
- All four sides are congruent.
- The diagonals bisect each other at right angles.
- The diagonals also bisect the angles of the rhombus.
The formulas for a rhombus's perimeter and area are essential for solving this problem. The perimeter is the sum of the lengths of all its sides, while the area can be calculated in a few ways, including using the side length and the height.
Step 1: Calculating the Side Length from the Perimeter
The perimeter of any polygon is the total length of its boundary. For a rhombus, since all four sides are equal, the perimeter (Pcap P
𝑃
) is simply four times the length of one side (ss
𝑠
). The formula is:
P=4scap P equals 4 s
𝑃=4𝑠
Given that the perimeter of the rhombus is 100 cm, we can substitute this value into the formula to find the side length.
100 cm=4s100 cm equals 4 s
100cm=4𝑠
To solve for ss
𝑠
, we divide the perimeter by 4:
s=100 cm4s equals the fraction with numerator 100 cm and denominator 4 end-fraction
𝑠=100cm4
s=25 cms equals 25 cm
𝑠=25cm
This means each side of the rhombus has a length of 25 cm.
Step 2: Calculating the Height from the Area and Side Length
The area (Acap A
𝐴
) of a rhombus can be thought of similarly to the area of a parallelogram. It is the product of its base (which is the side length, ss
𝑠
) and its corresponding height (hh
ℎ
). The height is the perpendicular distance between two opposite sides. The formula for the area in this context is:
A=s×hcap A equals s cross h
𝐴=𝑠×ℎ
We are given that the area is 175 cm² and we have calculated the side length (ss
𝑠
) to be 25 cm. We can now substitute these values into the area formula to find the height (hh
ℎ
).
175 cm2=25 cm×h175 cm squared equals 25 cm cross h
175cm2=25cm×ℎ
To isolate the height (hh
ℎ
), we divide the area by the side length:
h=175 cm225 cmh equals the fraction with numerator 175 cm squared and denominator 25 cm end-fraction
ℎ=175cm225cm
h=7 cmh equals 7 cm
ℎ=7cm
Conclusion
By first using the perimeter to determine the side length of the rhombus and then using that side length along with the given area, we can directly calculate the height. The logical progression from the perimeter to the side length and then to the height demonstrates the interrelationship between the fundamental properties of this geometric shape. The height of the rhombus is 7 cm.
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