Unit 6 Similar Triangles Homework 1 Ratio & Proportion

You're staring at the worksheet. "Unit 6: Similar Triangles. Homework 1: Ratio & Proportion." The title sounds straightforward — ratios, proportions, cross-multiplying, stuff you've seen before. But then you hit problem 3 and the triangles aren't drawn to scale, the variables are buried inside side lengths, and suddenly you're not sure if you're setting up the fraction correctly or if you just flipped the wrong pair.

Been there. This homework is where the unit lives or dies. Practically speaking, get the ratio mechanics solid now, and the rest of similar triangles — proofs, indirect measurement, coordinate geometry — clicks. Skimp here, and you're guessing on every problem that follows.

Let's walk through what this assignment actually tests, where students trip up, and how to make the setup automatic.

What This Homework Actually Covers

Unit 6 usually kicks off with ratio and proportion because similarity is proportional reasoning. Two figures are similar when their corresponding angles are congruent and their corresponding sides are proportional. That second part — the proportional sides — is exactly what Homework 1 drills.

You'll see three main problem types:

Straight proportion solving.
Something like x/12 = 8/18 or (3x + 2)/15 = 10/25. Cross-multiply, solve, check. These are the gimmes — but they're also where arithmetic errors hide.

Segment lengths with variables.
A triangle side labeled 2x + 3, the corresponding side on the similar triangle is 21, and the scale factor is given or implied. You set up the proportion, solve for x, then plug back to find the actual side length.

Multi-step problems with extended ratios.
"The sides of a triangle are in the ratio 3:4:5. The perimeter is 48. Find the side lengths." Or: "Two similar triangles have perimeters in the ratio 2:3. The area of the smaller is 18. Find the area of the larger." (Spoiler: area ratio is the square of the linear ratio. That one catches everyone the first time.)

The Vocabulary You Need Cold

Ratio — a comparison of two quantities, usually written as a fraction or with a colon (a:b or a/b).
Proportion — an equation stating two ratios are equal (a/b = c/d).
Cross products — in a/b = c/d, the products ad and bc are equal. This is the engine that solves proportions.
Scale factor — the multiplier that takes one figure to the similar figure. If triangle A → triangle B has scale factor 2, every side of B is twice the corresponding side of A.
Extended ratio — three or more quantities compared (a:b:c). Shows up in perimeter and angle problems.

Why Ratio & Proportion Feels Harder Than It Should Be

The algebra isn't new. Worth adding: you've solved proportions since pre-algebra. What makes this homework feel different is the geometry wrapper*.

In a pure algebra class, the proportion is given: x/5 = 12/20. Practically speaking, in geometry, you have to build* the proportion from a diagram. But you decide which side corresponds to which. So naturally, you decide which triangle goes in the numerator. You decide whether the scale factor is >1 or <1. That decision-making is where points vanish.

And the diagrams? But the 30° angle might look like 60°. The tiny triangle might be the bigger one. They're rarely drawn to scale. You cannot trust your eyes — you have to trust the markings (tick marks for congruent sides, arcs for congruent angles) and the similarity statement (△ABC ~ △DEF tells you A↔D, B↔E, C↔F).

How to Set Up Every Proportion Without Guessing

Here's the routine that works every time. Do it mechanically until it's automatic.

Step 1: Write the similarity statement if it's not given

If the problem says "△ABC is similar to △DEF," write it down: △ABC ~ △DEF. Think about it: the order matters*. The first letter corresponds to the first, second to second, third to third.

If the problem gives a diagram without a statement, create one* based on the markings. Match the angles with the same number of arcs. Match the sides with the same number of tick marks. Write your own similarity statement. This single step prevents 80% of correspondence errors.

Step 2: List corresponding sides in a table

Triangle 1	Triangle 2
AB	DE
BC	EF
CA	FD

Do this even for simple problems. In practice, it forces you to see the pairs. If a side length is an expression (2x + 3), write the expression in the table.

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Step 3: Pick two complete* pairs

You need one pair where you know both* side lengths. That gives you the scale factor. Worth adding: if you know AB = 6 and DE = 9, the scale factor from △ABC to △DEF is 9/6 = 3/2 = 1. 5.

If no pair is complete, you'll need to use the perimeter or an extended ratio — more on that below.

Step 4: Write the proportion with the known* pair as one ratio

Two valid setups (pick one and stay consistent):

Option A: Small over big
known small / known big = unknown small / unknown big

Option B: Triangle 1 over Triangle 2
AB / DE = BC / EF = CA / FD

Option B is safer because the similarity statement locks the order. You never have to think "wait, which one is numerator?" — the statement decides.

Step 5: Cross-multiply and solve

AB × EF = DE × BC
Plug in numbers. Solve for the variable. If the variable is inside an expression (2x + 3), solve for x first*, then substitute back to find the side length. Don't stop at x = 4 — the question probably asks for the side length, which is 2(4) + 3 = 11.

Step 6: Check against the scale factor

If your scale factor is 1.5, every side in the second triangle should be 1.5× the corresponding side in the first. Quick mental check: 6 → 9 ✓, 8 → 12 ✓, 10 → 15 ✓. If one comes out 14, you swapped a pair.

Extended Ratios: The Perimeter and Angle Problems

These show up on every version of this homework. Two flavors:

Flavor 1: Side lengths in extended ratio

"The sides of a triangle are in the ratio 5:12:13. The perimeter is 90. Find the side lengths. That's the whole idea.

The method:
Let the sides be 5x, 12x, 13x.
Perimeter = 5x + 12x + 13x = 30x = 90 → x = 3.
Sides: 15, 36, 39.

Why this works: The ratio gives the relative* sizes. The perimeter gives the absolute* scale. The variable x is the scale factor.

Flavor 2: Angle measures in extended ratio

"The angles of a triangle are in the ratio 2:3:

Find the measure of each angle."

The method:
Remember the Golden Rule of Triangles: The sum of the interior angles is always 180°.
Let the angles be $2x$, $3x$, and $4x$.
$2x + 3x + 4x = 180 \rightarrow 9x = 180 \rightarrow x = 20$.
Angles: $2(20) = 40^\circ$, $3(20) = 60^\circ$, $4(20) = 80^\circ$.

The Trap:
Students often try to use similarity ratios for angles. Stop. Similarity means angles are congruent* (equal), not proportional. If two triangles are similar, their angles are the same, while their sides are proportional. Never set up a ratio like $\frac{\text{Angle A}}{\text{Angle D}} = \text{scale factor}$. This is the fastest way to fail a similarity problem.

Summary Checklist for Success

The moment you face a similarity problem on a test, run through this mental checklist to ensure you haven't missed a step:

Check for Similarity first: Did you use AA, SSS, or SAS to prove they are actually similar?
The Statement Test: Did you write your similarity statement (e.g., $\triangle ABC \sim \triangle DEF$) based on matching arcs and tick marks?
The Table Test: Did you list your corresponding sides in a table to prevent "flipping" your fractions?
The Variable Test: If you solved for $x$, did you plug it back in to find the actual side length requested?
The Reality Check: Does the answer make sense? If the second triangle is larger than the first, your calculated side must also be larger.

By following this systematic approach, you move away from "guessing which number goes where" and toward a mathematical process that works every single time. Because of that, similarity isn't about memorizing formulas; it's about maintaining the relationship between parts. Master the ratio, and you master the geometry.