To solve this problem, we will use the formula for the area of a triangle, which is:
[tex]\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \][/tex]
Since we're given that the area of the triangle is 1,440 cm² and that the base is 5 times the height, we can set our variables as follows:
Let [tex]\( h \)[/tex] be the height of the triangle.
Therefore, the base [tex]\( b \)[/tex] of the triangle is [tex]\( 5h \)[/tex].
We can now express the area in terms of [tex]\( h \)[/tex]:
[tex]\[ 1440 = \frac{1}{2} \times (5h) \times h \][/tex]
Now we simplify and solve for [tex]\( h \)[/tex]:
[tex]\[ 1440 = \frac{1}{2} \times 5h^2 \][/tex]
[tex]\[ 1440 = \frac{5}{2}h^2 \][/tex]
Multiply both sides by 2 to get rid of the fraction:
[tex]\[ 1440 \times 2 = 5h^2 \][/tex]
[tex]\[ 2880 = 5h^2 \][/tex]
Now divide both sides by 5:
[tex]\[ \frac{2880}{5} = h^2 \][/tex]
[tex]\[ 576 = h^2 \][/tex]
To find [tex]\( h \)[/tex], we take the square root of both sides:
[tex]\[ h = \sqrt{576} \][/tex]
The square root of 576 is 24. Therefore:
[tex]\[ h = 24 \text{ cm} \][/tex]
The height of the triangle is 24 cm.
