Finding Unknown Side Lengths in Similar Polygons

Hello. In the previous lesson, you found the scale factor between similar polygons by dividing a target side length by its corresponding starting side length. Now we use that multiplier for the practical goal: finding a missing side length.

By the end of this lesson, you should be able to look at two similar polygons, match the correct sides, choose a known corresponding pair, and calculate an unknown length. You will also handle a common quiz variation: when similarity gives you a whole side, but the question asks for just one piece of it.

Plan for about 35–40 minutes: read and work through the examples carefully, watch the short worked example, then finish with the closed-note routine.

---

From a scale factor to a missing length

Similar polygons keep the same shape, so every corresponding side is changed by the same multiplier.

Suppose side $a$ in the starting polygon corresponds to side $a^{\prime }$ in the target polygon. If the scale factor from starting to target is $k$, then:

$$a^{\prime }=k\cdot a$$

To find an unknown side length, use this reliable routine:

1. Identify the two similar polygons or triangles.
2. Match corresponding vertices and sides. Use a similarity statement, angle markings, parallel lines, or matching endpoints—not how long a side looks.
3. Choose a known pair of corresponding sides and calculate the scale factor in the direction you need.
4. Multiply the known corresponding side by that factor to find the unknown.
5. Check whether the answer’s size makes sense. A reduction should produce a shorter length; an enlargement should produce a longer one.

You can also solve with a proportion. These are the same idea written differently:

$$\frac{\text{target side}}{\text{starting side}}=\frac{\text{target matching side}}{\text{starting matching side}}$$

The main rule is: keep the order consistent across both fractions.

For example, if a small triangle has a side of $6$, its corresponding side in a larger triangle is $9$, and another small side is $4$, then the scale factor from small to large is:

$$k=\frac{9}{6}=\frac{3}{2}$$

The corresponding unknown large side is:

$$\frac{3}{2}\cdot 4=6$$

Equivalently:

$$\frac{9}{6}=\frac{x}{4}$$ $$6x=36$$ $$x=6$$

The scale-factor method is usually faster when you have already identified the direction. The proportion method can be useful when you prefer to see both corresponding pairs lined up.

---

Example 1: Find a corresponding missing side

The first diagram below contains two triangles formed by intersecting lines. The arrow markings show that $\overline{AB}$ is parallel to $\overline{DE}$.

In the left diagram, consider:

$$△ABC$$

and

$$△EDC$$

The matching vertices are:

$$A\leftrightarrow E$$ $$B\leftrightarrow D$$ $$C\leftrightarrow C$$

So the corresponding sides include:

$$BC\leftrightarrow CD$$

and

$$AC\leftrightarrow CE$$

This matching step is more important than the arithmetic. In particular, $AC$ corresponds to $CE$, because both run from the intersection point $C$ to the vertex on the parallel line.

We know:

$$BC=5$$ $$CD=3$$

The scale factor from $△ABC$ to $△EDC$ is:

$$k=\frac{CD}{BC}$$ $$k=\frac{3}{5}$$

Since $CE$ corresponds to $AC=4$, multiply $4$ by the same reduction factor:

$$CE=\frac{3}{5}\cdot 4$$ $$CE=\frac{12}{5}$$ $$\boxed{CE=2.4}$$

A proportion gives the same result:

$$\frac{CE}{AC}=\frac{CD}{BC}$$ $$\frac{CE}{4}=\frac{3}{5}$$ $$CE=\frac{12}{5}$$

The answer is less than $4$, which fits the diagram’s relationships: $△EDC$ is a reduced version of $△ABC$.

Keep correspondence visible

When writing a proportion, write the side labels before replacing them with numbers:

$$\frac{CE}{AC}=\frac{CD}{BC}$$

Only then substitute:

$$\frac{CE}{4}=\frac{3}{5}$$

This small habit prevents a frequent error: using correct numbers from the diagram but pairing sides that do not correspond.

---

Example 2: When the unknown is only part of a side

The right-hand triangle diagram adds one extra step. Since $\overline{BD}$ is parallel to $\overline{AE}$, the smaller triangle and larger triangle are similar:

$$△CBD\sim △CAE$$

Their corresponding vertices are:

$$C\leftrightarrow C$$ $$B\leftrightarrow A$$ $$D\leftrightarrow E$$

Therefore:

$$CB\leftrightarrow CA$$ $$CD\leftrightarrow CE$$

Notice the second pair carefully:

$$CD\leftrightarrow CE$$

The side $CD$ in the small triangle corresponds to the entire side $CE$ in the large triangle. It does not correspond to $DE$, which is only one part of that larger side.

First find the full length $CA$. The diagram gives:

$$AB=3$$ $$BC=5$$

Since $B$ lies between $A$ and $C$:

$$CA=AB+BC$$ $$CA=3+5$$ $$CA=8$$

Now calculate the scale factor from small triangle $CBD$ to large triangle $CAE$:

$$k=\frac{CA}{CB}$$ $$k=\frac{8}{5}$$

Use it on the corresponding side $CD=4$:

$$CE=\frac{8}{5}\cdot 4$$ $$CE=\frac{32}{5}$$

At this stage, do not stop. The question asks for $DE$, not $CE$.

The complete side is made of the two pieces $CD$ and $DE$:

$$CE=CD+DE$$

So:

$$DE=CE-CD$$ $$DE=\frac{32}{5}-4$$

Rewrite $4$ with denominator $5$:

$$DE=\frac{32}{5}-\frac{20}{5}$$ $$\boxed{DE=\frac{12}{5}}$$

or:

$$\boxed{DE=2.4}$$

This kind of question tests two skills together:

• use similar triangles to find a missing whole corresponding side;
• use segment addition or subtraction to find the requested part.

---

Choose the method that matches the question

Here is a quick decision guide for similar-polygon side problems.

What you know	Best first move
A known corresponding pair and an unknown corresponding side	Find the scale factor, then multiply.
An unknown side in the starting figure but a known side in the target figure	Use the reciprocal scale factor, or divide by the scale factor.
The unknown is a piece of a larger side	Use similarity to find the whole corresponding side first, then add or subtract segments.
The diagram is confusing	Write the matching vertices and side pairs before using numbers.

For a missing side in the starting figure, division may be more natural. Suppose the factor from small to large is $\frac{3}{2}$, and the large corresponding side is $18$. Then the small side $x$ satisfies:

$$18=\frac{3}{2}x$$

You can divide by $\frac{3}{2}$:

$$x=18÷\frac{3}{2}$$ $$x=18\cdot \frac{2}{3}$$ $$x=12$$

Or reverse the scale factor first. From large to small, it is $\frac{2}{3}$:

$$x=\frac{2}{3}\cdot 18=12$$

Both approaches work because reversing a scale factor gives its reciprocal.

---

A quiz-ready solution format

On a quiz, avoid jumping immediately to a calculation. A clear solution can be only a few lines:

$$△ABC\sim △DEF$$ $$AB\leftrightarrow DE,AC\leftrightarrow DF$$ $$k=\frac{DE}{AB}=\frac{15}{10}=\frac{3}{2}$$ $$DF=\frac{3}{2}\cdot AC=\frac{3}{2}\cdot 8=12$$ $$\boxed{DF=12}$$

This format makes your reasoning visible. It also makes mistakes easier to locate:

• If the side labels do not match, it is a correspondence error.
• If matching sides are correct but the fraction has the wrong direction, it is a setup error.
• If the setup is correct but multiplication or simplification is wrong, it is a calculation error.
• If you find an entire side when the problem asks for a segment, it is a whole-versus-part error.

For a short new way to study this skill, use a retrieve-check-correct cycle:

1. Cover the worked examples and leave only the diagram visible.
2. Write the two triangles that are similar and list their matching side pairs.
3. Choose a scale factor direction and calculate the unknown.
4. Reveal the worked solution and compare each line, not just your final answer.
5. Record the type of any mismatch so your next review targets the real issue.

---

Key takeaways

To calculate an unknown side length in similar polygons:

1. Match corresponding vertices and sides.
2. Find the scale factor using a known corresponding pair.
3. Multiply or divide to obtain the unknown corresponding side.
4. If the question asks for part of a side, find the whole side first and then use segment addition or subtraction.
5. Check that the answer has a sensible size for an enlargement or reduction.

In the diagram examples:

$$CE=\frac{12}{5}=2.4$$ $$DE=\frac{12}{5}=2.4$$

Next, you will learn how to inspect an incorrect similar-polygon solution and decide whether the mistake came from correspondence, setup, or calculation.