Museum Angles
by Zack Kroll
As a fan of art and traveling I have been to some amazing museums with wonderful artwork in both the United States and in Europe. While walking around these museums I frequently ask myself, "where should I stand in order to view the painting in the most appropriate way".
We are looking at a 4ft by 4ft painting that is hanging on the wall. The bottom edge of the painting is 2ft above eye level. Our goal is to determine how far back we should stand, directly in front of the picture, in order to view it under the maximum angle?
To explore this situation properly we construct a replica of this sketch, but put it on a coordinate grid. This assists in measuring the segments of the figure. For us to make sure that we are choosing an appropriate distance to stand at, without going over the maximum, we need to see what that maxmimum angle is.
We place points A and B on the y-axis in accordance with the distance given in the problem and point C on the x-axis. After sliding point C we observe that the < ACB reaches approximately 30 degrees before it begins to decrease again.
When we constuct a circle O with the point A, B, and C, it appears that segment AB and AC are of equal length.
Unfortunately we cannot determine this solely based on our own sight and we must prove this somehow. According to the Central Angle Theorem, < ABC will be half the measure of < AOB. We measured < ACB at 30 degrees, so we should expect <AOB to be 60 degrees.
Triangle AOB is made up of three sides that all appear to be equal, meaning that it is an equilateral triangle. Once again we cannot assume this and must show whether is actually is. Segment OA and OB are both radii and therefore are equivalent. With <AOB = 60 degrees it means that the other two degrees must sum to 120. Because the side lengths opposite <OAB and <OBA are equal it means that the measure of the angles must also be equal. This tells us that in fact triangle AOB is an equilateral triangle.
Using the perpendicular line we constructed earlier to find the center of our circle O, we create a segment OD. The length of OA is 4 feet (equilateral triangle) and the length AD is 2 feet (half of AB). To find the measure of this side OD we use the Pythagorean Theorem and determine that it is approximately 3.46 feet.
Now that we know the length of OD we construct another perpendicular line from that segment. The point C lies directly on that perpendicular line and therefore is equidistant from the y-axis as O.
We can conclude that the appropriate distance to stand from the painting is 3.46 feet.
