Answer:
To find the greater height of a triangle, we first need to understand which side this height corresponds to. In this triangle, the greater height will be the height relative to the longest side as a base, since it will yield the largest area for the same perimeter.
The given sides of the triangle are 5 cm, 5 cm, and 8 cm. Since the triangle has two sides of equal length (5 cm), and one side longer (8 cm), the greater height would be the height dropped from the vertex opposite the longest side, which is 8 cm in this case.
To calculate the height, we can use Heron's formula to find the area \( A \) of the triangle first, and then use the area to find the height \( h \). The formula for the area \( A \) of a triangle with sides \( a \), \( b \), and \( c \) is:
[tex]\[A = \sqrt{s(s-a)(s-b)(s-c)}\][/tex]
where \( s \) is the semiperimeter of the triangle:
[tex]\[s = \frac{a + b + c}{2}\][/tex]
In this case:
- \( a = 5 \) cm,
- \( b = 5 \) cm,
- \( c = 8 \) cm.
First, let's calculate the semiperimeter \( s \):
[tex]\[s = \frac{5 + 5 + 8}{2}\][/tex]
Then, we can use this to find the area \( A \). Once we have the area, the height \( h \) relative to the side of length 8 cm is given by:
[tex]\[h = \frac{2A}{c}\][/tex]
Let's perform these calculations.
The area of the triangle is [tex]\(12 \text{ cm}^2\)[/tex] and the greater height, which is the height relative to the base of 8 cm, is [tex]\(3 \text{ cm}\)[/tex]. Thus, the greater height of the triangle is 3 cm.
Step-by-step explanation: