Question

Architecture Maria is visiting Washington, D.C. She wants to know the height of the Washington Monument. The monument's shadow is 111 feet at the same time that Maria's shadow is 1 foot. Maria is 5 feet tall. a. Draw a figure to represent the problem. Label all known distances. b. Outline two similar triangles in red. c. Determine the height of the Washington Monument.

Short answer

The Washington Monument is 555 feet tall.

Step-by-step solution

Sketch the Figure

First, draw a simple diagram to represent the problem. Start by sketching the Washington Monument as a tall vertical line with a shadow extending horizontally. Beside it, draw Maria as a shorter vertical line with a corresponding shorter shadow. Label Maria's height as 5 feet and her shadow as 1 foot. Label the shadow of the monument as 111 feet.

Identify the Similar Triangles

In the diagram, observe that both Maria and the Washington Monument create right-angled triangles with their shadows. These triangles are similar because they both share the angle with the horizon (the sun’s angle is the same for both). Highlight these two triangles in red to clearly indicate their similarity.

Set Up the Proportion Equation

Use the properties of similar triangles to set up a proportion. For similar triangles, the ratio of corresponding sides is equal. Thus, we can write: \( \frac{\text{Height of Maria}}{\text{Shadow of Maria}} = \frac{\text{Height of Monument}}{\text{Shadow of Monument}} \). Substitute the known values: \( \frac{5}{1} = \frac{h}{111} \).

Solve the Equation for the Monument's Height

To find the height 'h' of the monument, solve the proportion equation \( \frac{5}{1} = \frac{h}{111} \).Cross multiply to get: \( 5 \times 111 = 1 \times h \), which simplifies to \( h = 555 \). Thus, the height of the Washington Monument is 555 feet.

Key concepts

Similar Triangles

In geometry, similar triangles are triangles that have the same shape but possibly different sizes. This means that their corresponding angles are equal, and their corresponding sides are proportional. To determine the similarity of triangles, like in the case of Maria and the Washington Monument, examine the angles involved.

• If two triangles have two pairs of corresponding angles equal, then the triangles are similar.
• Similar triangles are often used to deduce unknown measurements, as they maintain a constant ratio between corresponding sides.

In the exercise provided, both Maria and the monument form right-angled triangles with their shadows. These triangles are similar because they share the angle at which the sun's rays strike them, making it possible for us to use proportions to calculate the monument's height.

Proportion Method

The proportion method involves using a ratio to compare the sizes of different structures or figures in similar triangles. When you have two similar triangles, their corresponding sides are proportional to each other. This gives us a powerful tool to calculate unknown lengths.

• Set up a proportion using a known length and a corresponding unknown length. Using ratios helps to maintain the balance between the two sets of triangles.
• The formula used for such problems can be expressed as: \( \frac{\text{Side 1}}{\text{Side 2}} = \frac{\text{Side 3}}{\text{Side 4}} \).

In our situation with Maria and the Washington Monument, applying the proportion \( \frac{5}{1} = \frac{h}{111} \) allows us to solve for the height of the monument swiftly by cross-multiplying.

Real-life Applications

Understanding similar triangles and proportions doesn't just stay within the bounds of textbooks. In the real world, these concepts can be applied to a variety of practical problems, such as estimating the height of tall structures or even measuring distances that are hard to reach.

• Architects and engineers use these geometrical principles to design and verify the stability of various structures.
• Surveyors frequently use similar triangles in determining the height of remote objects when direct measurement is not feasible.

This practical application of geometry makes mastering similar triangles and proportions immensely valuable, allowing estimation and measurements with accuracy and ease.

Geometry Concepts

Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. It offers several foundational concepts, such as lines, angles, and triangles which are essential for problem-solving in various fields.

• Triangles are a fundamental shape in geometry, characterized by having three sides and three angles. Understanding their properties enables us to solve more complex problems.
• Properties of right-angled triangles, including the Pythagorean theorem, are central to many geometry problems.

Through geometry, students learn to visualize and solve problems through diagrams, like drawing similar triangles to solve for unknown heights, highlighting the way geometric concepts can be applied creatively and accurately in both academic and practical contexts.