Q. 100 - Precalculus Enhanced with Graphing Utilities

Question

An equilateral triangle is one in which all three sides are of equal length. If two vertices of an equilateral triangle are (0, 4) and (0, 0), find the third vertex. How many of these triangles are possible?

Step-by-Step Solution

Answer

Third vertex is (2sqrt(3), 2) or (-2sqrt(3), 2). Two triangles are possible.

Step 1: Find Side Length

Distance between \((0,4)\) and \((0,0)\) is 4. All sides must equal 4.

Step 2: Find Third Vertex

Let third vertex be \((x,y)\). From \(x^2+y^2=16\) and \(x^2+(y-4)^2=16\), we get \(y=2\) and \(x=\pm 2\sqrt{3}\).

Step 3: Answer

Third vertex: \((2\sqrt{3}, 2)\) or \((-2\sqrt{3}, 2)\). Two triangles are possible.