can u show me how to do the problem

Question

An equilateral triangle has vertices at (0, 0), (8, 0), and (4, y), where y > 0. What is the value of y? Round to the nearest hundredth

Answer 1

Start by drawing this on graph paper.  In geometry, especially coordinate geometry, it always helps to see a picture of what you're trying to solve.

 

Whether you draw it or not, you can see that that length of the base, the line segment from (0,0) to (8,0) is 8.  So the other two sides also have to be 8.

 

What's the length of the side from (0,0) to (4,y)?  By the formula for the distance between two points, it's:

 

√[(4 - 0)² + (y - 0)²] = √[4² + y²] = √[16 + y²]. 

 

You know that has to be 8, so set up and solve the equation:

 

√[16 + y²] = 8

16 + y² = 8² = 64

y² = 48

y = ±√48 = ±√(16 * 3) = ± 4√3

 

Since they told you that y > 0, you know that y has to be 4√3.

 

Calculate this to the nearest hundredth, and you get 6.93.

 

You can check this by finding the distance from (4, 4√3) to the other point (8,0).  You will see that it is also 8.