The diagonals of the rhombus are 6 cm and 8 cm. Determine the height of the diamond.

1) Find the area of ​​the rhombus.
We know the lengths of the diagonals of the rhombus. We can find the area of ​​the rhombus. Let’s write the formula for the area:
S = 1/2 × d1 × d2,
where d1 and d2 are rhombus diagonals.
Substituting the values ​​of the diagonals in the formula, we find the area:
S = 1/2 × 6 × 8 = 24 cm².
2) Find the side of the rhombus.
Based on the properties of the rhombus, knowing the diagonals, we can find its side. Let’s write down:
The sum of the squares of the diagonals is equal to the square of the side times 4:
d1² + d2² = 4a²,
where d1 and d2 are the diagonals of the rhombus, and is the side of the rhombus.
Substituting the values ​​of the diagonals in the formula, we find the side of the rhombus:
6² + 8² = 4 × a²,
4 × a² = 36 + 64,
4 × a² = 100,
a² = √ (100: 4),
a² = √25,
a = 5 cm.
3) Find the height of the rhombus. Let’s write the formula for the area through the side of the rhombus and the height:
S = ah,
where a is the side of the rhombus, h is the height.
Let us express the height from this formula:
h = S: a.
We found the area and side of the rhombus, which means:
h = 24: 5 = 4.8 cm.
Answer: The height of the rhombus is 4.8 cm.