Two vertices of an equilateral triangle are located at the points (\frac{1}{2}, 2) and (\f… — ACT

Two vertices of an equilateral triangle are located at the points $(\frac{1}{2}, 2)$ and $(\frac{5}{2}, 2)$ in the standard $(x,y)$ coordinate plane. If the third vertex is located above the given line segment, what are its coordinates?

A
$(\frac{3}{4}, 2 + \sqrt{3})$
$(\frac{3}{2}, 2 + \sqrt{3})$ Answer
C
$(\frac{7}{4}, 2 + \sqrt{3})$
D
$(\frac{3}{2}, \sqrt{7})$
E
$(\frac{3}{2}, 4)$

Answer

(\frac{3}{2}, 2 + \sqrt{3})

The midpoint of the base segment is at

(\frac{3}{2}, 2)

and the vertical height of the equilateral triangle is

\sqrt{3}

. Since the vertex lies above the segment, we add the height to the

y

-coordinate of the midpoint, giving the coordinates

(\frac{3}{2}, 2 + \sqrt{3})

Step-by-Step Solution

Find the midpoint and length of the segment connecting the two given vertices

(\frac{1}{2}, 2)

and

(\frac{5}{2}, 2)

Since both points lie on the horizontal line

y = 2

, the distance between them is

\frac{5}{2} - \frac{1}{2} = 2

. The

x

-coordinate of the midpoint is

\frac{\frac{1}{2} + \frac{5}{2}}{2} = \frac{3}{2}

, so the midpoint is at

(\frac{3}{2}, 2)

The third vertex of an equilateral triangle lies on the perpendicular bisector of the opposite side, which passes through its midpoint.

Calculate the height of the equilateral triangle using the Pythagorean theorem.

The side length of the triangle is

2

. The distance from a vertex to the midpoint of the opposite side is

1

. The height

h

satisfies

1^2 + h^2 = 2^2 \Rightarrow 1 + h^2 = 4 \Rightarrow h = \sqrt{3}

The height of an equilateral triangle divides it into two

30^\circ\text{-}60^\circ\text{-}90^\circ

right triangles.

Determine the coordinates of the third vertex by applying the height to the midpoint.

Since the base is horizontal, the altitude is vertical. Thus, the third vertex has the same

x

-coordinate as the midpoint,

\frac{3}{2}

. The

y

-coordinate is the

y

-coordinate of the midpoint plus the height, which is

2 + \sqrt{3}

The vertex must be located above the segment, so we add the height to the

y

-coordinate of the midpoint.

Key Concept

Distance and Midpoint Formulas

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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