Question

Suppose one end of an eight-foot straight pole is four feet from the center of projection. Moreover, suppose that a straight line from the center of projection to one end of the pole intersects the projection plane at a point that is one foot from the center of projection. If the pole is parallel to the projection plane, how long is the image of the pole in the projection plane?

Short answer

The image of the pole is 2 feet long.

Step-by-step solution

Understanding the Problem

We have an eight-foot pole parallel to the projection plane. One of its ends is 4 feet away from the center of projection, and the projection line intersects the plane at 1 foot from the center of projection. We need to find the length of the pole's image on the projection plane.

Setting Up Similar Triangles

The situation forms similar triangles. The small triangle has sides of 1 unit (distance from center to intersection) and the unknown image length, and the large triangle has sides of 4 units (distance from center to pole) and 8 units (actual pole length).

Applying the Similar Triangles Ratio

The ratio of the lengths of corresponding sides in similar triangles is equal. Therefore, we set up the proportion: \( \frac{1}{x} = \frac{4}{8} \) where \( x \) is the length of the pole's image on the projection plane.

Solving the Proportion

Solving the equation \( \frac{1}{x} = \frac{1}{2} \), we cross-multiply to get \( 2 = x \). This means the length of the image is 2 feet.

Key concepts

Projection

When thinking about projection in geometry, imagine casting a shadow. Much like how the sun casts shadows on the ground, a central point (the center of projection) projects a figure onto a different plane. In the exercise, the pole acts similarly to an object casting a shadow. One end of the pole is 4 feet from this center of projection. The projection plane, which is like the surface capturing the shadow, experiences the image of the pole shortening because of the angle at which the light (or projection line) hits it. The center of projection to the plane is only 1 foot at the point of intersection, making the projection shorter than the actual pole. This concept is crucial in understanding how dimensions change from reality into a projected image.

Similar Triangles

A key geometric property in this scenario is the concept of similar triangles. Similar triangles are triangles that, while different in size, have equal corresponding angles and proportional sides. In the problem, two triangles are formed: the first large triangle has the actual pole length of 8 feet and a distance from the projection center of 4 feet. There is also a smaller triangle with a base of 1 foot. Both triangles share an angle from the projection center, making them similar. This similarity is what allows us to calculate the length of the pole’s image by comparing the proportional lengths of their sides. Understanding similar triangles helps bridge the gap between different sizes of objects and their relational properties in geometry.

Proportions

Proportions allow us to express the equivalence between two ratios. This exercise shows how unknown lengths in geometry can be identified through proportions derived from similar triangles. The proportion set up from the similar triangles \(\frac{1}{x} = \frac{4}{8}\)relates the known distances to the unknown image length. When you cross-multiply, it helps isolating the unknown variable by equating the product of the means to the product of the extremes. Solving this equation \(2 = x\)yields the image length to be 2 feet. Proportional reasoning is a powerful tool not just in geometry, but in many areas where relationship comparisons are necessary. Through proportions, we learn how to solve for missing information by using given data in a structured mathematical framework.