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How You Use the Triangular Proportionality Theorem Every Day

Diagram showing the triangle proportionality theorem

The triangle proportionality theorem is a geometric law stating that when you draw a line parallel to one side of a triangle, it’ll intersect the other two sides of the triangle and divide them proportionally. Regardless of whether they're obtuse, acute, or right triangles, this theorem can be used to determine unknown lengths within similar triangles.

In the model above, you see sides of a triangle marked A, B, and C. The perpendicular line segment DE intersects both sides B and A, therefore making it a transversal. Since triangles EDC and ABC are two different sizes, they are not congruent. But since DE is parallel to BA you can postulate, or prove, that they're corresponding angles. According to the triangular proportionality theorem, you can determine the length from E to C by using this ratio:

CB:CA=CD:CE

Use the diagram below as a worksheet. When you plug in the numbers from the diagram, the ratio would be 8:10=5:?. You would then use fractions to solve for x and discover the unknown length:

triangle proportionality theorem: Formula for the unknown length of one side of a triangle

Triangle Proportionality Theorem in the Professional World

Diagram showing a mountain used as an example for the triangle proportionality theorem

In trades where constructing and forming multiple routes is needed, the triangle proportionality theorem is often used to determine how long a route should be. For example, say an engineer wants to build a mountain road. The lengths from the top are 900 feet and 800 feet, respectively. The engineer has determined they need to start the route at 300 feet and 525 feet below the peak. Use the Triangle Proportionality Theorem to determine the length of the route:

Formula for determining the length of the route

Using the Theorem in Construction

The triangle proportionality theorem is useful for construction. For example, say a contractor is building support beams for the roof of a house. One support beam has already been applied to the upper part of the room, but an additional one must be constructed. The measurements are as follows:

Picture of a roof with each support beam measurement is indicated

Since we know that the smaller support beam is 8 feet and the distance from the top of the roof to the first support beam is 7 feet, we can use the triangle proportionality theorem to figure out that the bottom support beam should be 6.2 feet.

Using the Triangle Proportionality Theorem in Daily Life

How often have you needed to know the length of something but didn’t have the tools or knowledge to figure out? For example, say you want to rearrange your rug so it’s hidden under your sectional couch and parallel to the nearby carpeted area. The dimensions are as follows:

Photo of a living room with measurement of the rug indicated

Using the triangle proportionality theorem, you get the ratio 9:12=4:x. When you solve for x, the result is 5.3 ft of the angled rug should be showing.

Your New Favorite Theorem

Using the triangle proportionality theorem, you can find out how long detour roads should be and how much wood you need to construct support beams. Not only will this rule help you in trigonometry — it’ll teach you to see life from a more practical, critical, and even safer perspective.

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