Answer:
24 units
Step-by-step explanation:
To find the length of the shortest altitude in a triangle, we can use Heron's formula to calculate the area of the triangle, and then use the formula for the altitude:
[tex]\boxed{\boxed{\sf \textsf{Area} = \sqrt{s(s-a)(s-b)(s-c)}}}[/tex]
where [tex] s[/tex] is the semi-perimeter of the triangle, and [tex] a[/tex], [tex] b[/tex], and [tex] c[/tex] are the lengths of the sides. Once you have the area, we can use the formula for the altitude:
[tex]\textsf{Altitude} = \dfrac{2 \times \textsf{Area}}{\textsf{Base}}[/tex]
In this case, the sides of the triangle are given as 30, 40, and 50. The semi-perimeter [tex] s[/tex] is calculated as:
[tex] s = \dfrac{30+40+50}{2} = 60[/tex]
Now, apply Heron's formula:
[tex]\textsf{Area} = \sqrt{60 \times (60-30) \times (60-40) \times (60-50)}[/tex]
[tex]\textsf{Area} = \sqrt{60 \times 30 \times 20 \times 10}[/tex]
[tex]\textsf{Area} = \sqrt{360000}[/tex]
[tex]\textsf{Area} = 600[/tex]
Now, use the formula for the altitude:
[tex]\textsf{Altitude} = \dfrac{2 \times 600}{\textsf{Base}}[/tex]
Since we need the shortest altitude, it means we need to keep the longest side as the base (50).
[tex]\textsf{Altitude} = \dfrac{2 \times 600}{50}[/tex]
[tex]\textsf{Altitude} = \dfrac{1200}{50}[/tex]
[tex]\textsf{Altitude} = 24[/tex]
Therefore, the length of the shortest altitude in the triangle is 24 units.
