## Proportional Parts in Triangles and Parallel Lines

Parallel lines may seem boring, but they have their uses. One of their uses appears in the Triangle Proportionality Theorem, which uses a line constructed parallel to one side of a triangle to establish proportions for the other two sides.

First, let's briefly cover parallel lines. Two lines are parallel if they do not intersect. Portions of those lines, such as rays and line segments, are also parallel. In the drawing below, line segment $WO$ is parallel to line segment $LF$:

## What is the Triangle Proportionality Theorem?

The triangle proportionality theorem states that if you draw a line constructed parallel to one side of a triangle intersects the other two sides of the triangle and divides the remaining two sides proportionally.

Let's break this down. Draw a triangle (scalene, right, obtuse -- it does not matter) with one side horizontal to you. Label it $△SLI$ with side $LI$ at the bottom, horizontal to you. Side $SL$ should be to your left and $SI$ to your right.

Construct a line parallel to $LI$ anywhere you like so it cuts across the interior of the triangle. Where the parallel line crosses sides $SL$ and $SI$, label those points $C$ and $E$.

Any way you slice it, $△SCE$ and $△SLI$ are proportional to each other. More, side $SL$ has been divided into two segments, $SC$ and $CL$, that are proportional to the two segments side $SI$ is divided into, $SE$ and $EI$. The Triangle Proportionality Theorem is your assurance these conclusions are true.

You can write ratios to show these proportions:

## Similar Proportional Triangles

Here is $△BOX$. Construct a line parallel to line segment (and side) $OX$. Where the parallel line crosses sides $BO$ and $BX$, label the points $E$ and $D$. Write two proportions you know.

Did you get two of these?

## Triangle Proportionality Theorem Examples

You can use this theorem to find the unknown lengths of portions of the sides of triangles if you know the other three portions. Suppose you are mapping out a delivery route, and you notice your five stops, , make a large triangle:

The highway department reports the road from $O$ to $U$ is closed due to bridge repairs. You know the road going from $T$ to $E$ is parallel to the $OU$ road.

You know the distance from your starting point, $O$, to $T$ is 6 km, and $TR$ is 15 km. You also know that the distance from stop $R$ to $E$ is 10 km, but you have no idea how far you will drive from $E$ to $U$.

Thanks to the Triangle Proportionality Theorem, you can easily calculate it. You know all this:

All you have to do is solve the proportions. You can use cross-multiplying and division, or you can multiply both sides times 10 to isolate $x$.

Cross-multiplying and division:

--OR--

Multiply both sides times 10 to isolate $x$:

Simplified, gives you 4 km. The distance from $E$ to $U$ is 4 km.

## Triangle Proportionality Theorem Practice

Here is a slightly deranged reason to apply the Triangle Proportionality Theorem, unless you are a zookeeper. Then it might be useful.

Your crocodile enclosure comprises two tall, parallel walls, with the near wall 10 m tall but still shorter than the far wall. You know the distance from one wall to the other is 12 meters (your crocodiles have lots of room).

The weather is turning cold so you plan to provide soothing overhead heat lamps for your crocodiles. You need to run aluminum tubes (to support the lamps) from the near, short wall to the taller, far wall. How long must the tubes be to reach across? If you cut them too short, they will drop into the enclosure.

### Right Triangle Proportions

Construct an imaginary triangle out from the crocodile enclosure's near wall. Go out 24 m and measure from the ground to the top of the near wall, or better yet, use the Pythagorean Theorem to calculate the hypotenuse of the right triangle without ever leaving your zookeeper office!

The ratios are now clear:

How long must your aluminum lighting tubes be to safely span the crocodile enclosure? Did you get the right answer? They must be 13 meters long.

## Lesson Summary

If you carefully read this lesson, studied the drawings, and watched the video, now you can describe and apply the Triangle Proportionality Theorem, which states that a line parallel to one side of triangle intersects the other two sides of the triangle and divides the remaining two sides proportionally. You can also use the Triangle Proportionality Theorem to find solutions to common situations encountered in daily life.

## What you learned:

After reading these directions, studying the drawings and watching the video, you will learn to:

• Describe and apply the Triangle Proportionality Theorem
• Use the Triangle Proportionality Theorem to common situations encountered in daily life
Instructor: Malcolm M.
Malcolm has a Master's Degree in education and holds four teaching certificates. He has been a public school teacher for 27 years, including 15 years as a mathematics teacher.

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