The perimeter of an equilateral triangle is 624 centimeters. The height of this triangle is k√3 centimeters, where k is a constant. What is the value of k?
The Correct Answer and Explanation is:
We are given that the perimeter of an equilateral triangle is 624 centimeters and the height of the triangle is expressed as k3k\sqrt{3}k3 centimeters, where kkk is a constant. We need to find the value of kkk.
Step 1: Understand the Properties of an Equilateral Triangle
In an equilateral triangle, all three sides are equal, and the height divides the triangle into two 30-60-90 right triangles. The relationship between the sides and the height is a key to solving this problem.
Let sss represent the side length of the equilateral triangle. The perimeter of an equilateral triangle is the sum of the lengths of all three sides, so:Perimeter=3s\text{Perimeter} = 3sPerimeter=3s
Given that the perimeter is 624 cm, we can solve for sss:3s=624⇒s=6243=208 cm3s = 624 \quad \Rightarrow \quad s = \frac{624}{3} = 208 \, \text{cm}3s=624⇒s=3624=208cm
Step 2: Relate the Height and the Side Length
The height hhh of an equilateral triangle can be found using the Pythagorean theorem in one of the right triangles formed by drawing an altitude. The altitude divides the equilateral triangle into two 30-60-90 triangles, where the ratios of the sides are 1: 3\sqrt{3}3: 2.
In a 30-60-90 triangle, the height hhh is related to the side length sss by the formula:h=s32h = \frac{s \sqrt{3}}{2}h=2s3
Substitute s=208s = 208s=208 into this equation:h=20832=1043h = \frac{208 \sqrt{3}}{2} = 104 \sqrt{3}h=22083=1043
Step 3: Compare the Given Height with the Expression k3k \sqrt{3}k3
We are told that the height of the triangle is k3k \sqrt{3}k3 centimeters. From the previous step, we found that the height is 1043104 \sqrt{3}1043 centimeters. Therefore, by comparing these two expressions, we can conclude that:k=104k = 104k=104
Thus, the value of kkk is 104\boxed{104}104.
