### Marianne Parsons

Using Geometer's Sketchpad, we can explore various types of problems.

A 4'-0"x4'-0" picture hangs on a wall such that its bottom edge is 2'-0" above eye level. How far back from the picture should you stand, directly in front of the picture, in order to view the picture under the maximum angle?

Let's begin by drawing a sketch of the problem. Let A be the point at the top of the picture, B be the point at the bottom of the picture, and line CE be along your eye level.

We can see from the above illustration that in order to find the distance from the picture, we must first examine the maximum angle. This is the marked as the angle AEB.

View an animation of the angle AEB as point E moves along the eye level line. Notice what happens to that angle as you move closer to, or farther from, the picture on the wall. When is the angle AEB at a maximum? You will need Geometer's Sketchpad to view this animation.

This animation shows us that the maximum angle of AEB is 30 degrees. If our point E moves closer the the picture, the angle will decrease. If our point moves farther from the picture, our angle will again decrease.

### How do we find the angle AEB geometrically?

In order to get the maximum angle for AEB, consider the entire triangle AEB.

Let's first consider the midpoint, M1, of our line BE. Draw the perpendicular bisector of BE.

Next, consider the midpoint, M2, of our line AE. Draw the perpendicular bisector of line AE.

Finally, consider the midpoint, M3, of our line AB. Draw the perpendicular bisector of AB.

By drawing the perpendicular bisectors through of these lines at their midpoints, we can see they intersect. We will call the point of intersection point P. Note, P is the circumcenter of our triangle AEB.

In order for angle AEB to be at a maximum, point E must line up with point P. Let's move our point E.

So what is the value of angle AEB?

Angle AEB = Angle BEC = Angle AEP = 30 degrees. How do we know? It is helpful to remember what other angles comprise angle CEP.

View the animation as point E is moved to line up with our point P.

### What is the distance of line segment CE?

Now we can use basic trigonometry operations to find the distance of the line segment CE, which we will call x. Consider the triangle formed by AEC.

Given:

We know the height of this triangle is the distance from the top of the picture to our eye level. The picture is 4'-0" in height, and sits 2'-0" above our eye level. So the height is 6'-0".

We have found the angle AEC to be the sum of angle AEB and angle BEC. Therefore angle AEC is 60 degrees.

Find:

Find the length of the base x of this triangle, also called segment CE.

Solve:

Recall from trigonometry- given an angle and opposite side length, the adjacent side can be found using the formula:

Theta is the angle we have found, 60. Opposite is the length of the side opposite our angle, 6'-0". Adjacent is the length of the side adjacent to our angle, x. Plug in the given values for our variables, and solve for x.

Multiply both sides by x, and cancel:

Divide both sides by tan 60, and cancel:

Compute:

Or approximately,

Therefore, if we were to stand approximately 3'-5 9/16" (3.464ft) from the face of the picture, we would be viewing that picture under the maximum angle.