Question

8 fill in the blank 1 point the side of an equilateral triangle has a length of 12. what is the altitude? altitude = choose your answer...

Explanation:

Step1: Recall the property of equilateral triangle

In an equilateral triangle, the altitude \( h \) divides the triangle into two right - angled triangles. If the side length of the equilateral triangle is \( a \), and we consider one of the right - angled triangles, the hypotenuse is \( a \), one leg is \( \frac{a}{2} \) (half of the side of the equilateral triangle) and the other leg is the altitude \( h \).
We can use the Pythagorean theorem \( c^{2}=a^{2}+b^{2} \) (where \( c \) is the hypotenuse and \( a,b \) are the legs of the right - triangle). Here, \( c = a=12 \), \( a_{1}=\frac{a}{2} = 6 \) and \( b = h \).

Step2: Apply the Pythagorean theorem

According to the Pythagorean theorem, \( h^{2}+(\frac{a}{2})^{2}=a^{2} \).
Substitute \( a = 12 \) into the formula:
\( h^{2}+6^{2}=12^{2} \)
\( h^{2}+36 = 144 \)
Subtract 36 from both sides: \( h^{2}=144 - 36=108 \)
Take the square root of both sides: \( h=\sqrt{108}=\sqrt{36\times3}=6\sqrt{3}\approx 10.392 \) (we can also keep it in exact form \( 6\sqrt{3} \))

Answer:

\( 6\sqrt{3} \) (or approximately \( 10.39 \))