Mark as brainlists
Answer:
To solve the problem, let's denote the lengths of the sides of the triangle as \( 2x \), \( 3x \), and \( 4x \) respectively, since they are in the ratio 2:3:4. Given that the perimeter of the triangle is 144 cm, we have the equation:
\[
2x + 3x + 4x = 144
\]
Simplify and solve for \( x \):
\[
9x = 144
\]
\[
x = \frac{144}{9} = 16
\]
Now, substitute \( x = 16 \) back into the expressions for the sides of the triangle:
\[
\text{Side lengths: } 2x = 32 \text{ cm}, \quad 3x = 48 \text{ cm}, \quad 4x = 64 \text{ cm}
\]
Now, to find the area of the triangle, we can use Heron's formula. First, calculate the semi-perimeter \( s \):
\[
s = \frac{2x + 3x + 4x}{2} = \frac{144}{2} = 72 \text{ cm}
\]
Now, apply Heron's formula for the area \( A \):
\[
A = \sqrt{s(s-a)(s-b)(s-c)}
\]
where \( a = 32 \), \( b = 48 \), \( c = 64 \). Substitute these values:
\[
A = \sqrt{72 \cdot (72 - 32) \cdot (72 - 48) \cdot (72 - 64)}
\]
Calculate each term:
\[
72 - 32 = 40, \quad 72 - 48 = 24, \quad 72 - 64 = 8
\]
\[
A = \sqrt{72 \cdot 40 \cdot 24 \cdot 8}
\]
Calculate step by step:
\[
40 \cdot 24 = 960, \quad 960 \cdot 8 = 7680
\]
\[
A = \sqrt{72 \cdot 7680}
\]
Now, compute \( 72 \cdot 7680 \):
\[
72 \cdot 7680 = 552960
\]
Finally, take the square root:
\[
A = \sqrt{552960} = 744 \text{ cm}^2
\]
So, the area of the triangle is \( 744 \) square centimeters.
Next, to find the height corresponding to the longest side (which is \( 64 \) cm), use the formula for the height \( h \) from the longest side:
\[
h = \frac{2 \cdot \text{Area}}{\text{Length of longest side}}
\]
Substitute the values we found:
\[
h = \frac{2 \cdot 744}{64} = \frac{1488}{64} = 23.25 \text{ cm}
\]
Therefore, the height corresponding to the longest side (64 cm) is \( \boxed{23.25} \) cm.
