Question

use the diagram of an equilateral triangle to answer the question.
what is the height of the triangle?
a. ( 5sqrt{2} ) cm
b. ( 5sqrt{3} ) cm
c. ( 10sqrt{2} ) cm
d. ( 10sqrt{3} ) cm

Explanation:

Step1: Analyze the properties of an equilateral triangle

In an equilateral triangle, the height divides the base into two equal parts. If the side length of the equilateral triangle is \(a = 10\space cm\), then the base of the right - triangle formed (half of the original base) is \(b=\frac{a}{2}=\frac{10}{2} = 5\space cm\), and the hypotenuse is \(c = 10\space cm\).

Step2: Apply the Pythagorean theorem

Let the height be \(h\). According to the Pythagorean theorem \(a^{2}=b^{2}+c^{2}\) (for a right - triangle, here \(c\) is the hypotenuse of the right - triangle, \(b\) is one of the legs, and \(h\) is the other leg). So \(h=\sqrt{c^{2}-b^{2}}\). Substitute \(c = 10\) and \(b = 5\) into the formula: \(h=\sqrt{10^{2}-5^{2}}=\sqrt{(10 + 5)(10 - 5)}=\sqrt{15\times5}=\sqrt{75}=5\sqrt{3}\space cm\).

Answer:

B. \(5\sqrt{3}\space cm\)