Similar Triangles: Finding Perimeter of ΔDEF When ΔABC Perimeter is 64

Q: How do I know which sides correspond to each other?

Look at the similarity statement $ \triangle DEF \sim \triangle ABC $. The order of vertices tells you: D↔A, E↔B, F↔C. So side EF corresponds to side BC, which are the sides given in the diagram.

Q: Why can I use any pair of corresponding sides to find the ratio?

In similar triangles, all corresponding sides have the same ratio. Whether you use $ \frac{DE}{AB} $, $ \frac{EF}{BC} $, or $ \frac{DF}{AC} $, you'll get the same ratio!

Q: Does the same ratio apply to perimeters?

Yes! If the side ratio is $ \frac{3}{4} $, then the perimeter ratio is also $ \frac{3}{4} $. This works because perimeter is just the sum of all sides, and each side has the same ratio.

Q: How can I tell which triangle is smaller from the diagram?

Compare the given side lengths: EF = 18 and BC = 24. Since 18

Q: What if I accidentally use the wrong ratio direction?

You'll get an impossible answer! If you use $ \frac{4}{3} \times 64 \approx 85.3 $, the smaller triangle would have a larger perimeter than the bigger triangle, which makes no sense. Always check that your answer is reasonable!

Similarity Ratios with Perimeter Applications

$ΔDEF∼Δ\text{ABC}$

If the perimeter of $ΔABC$ is 64, then what is the perimeter of $ΔDEF$ ?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the perimeter of the triangle according to the similarity ratio

00:03 The triangles are similar according to the given data

00:07 We want to find the similarity ratio

00:11 Let's substitute the side values according to the given data and solve

00:15 We'll factor each number with factor (6) and reduce

00:19 This is the similarity ratio between the triangles

00:23 The perimeter ratio equals the similarity ratio

00:29 Let's substitute appropriate values and solve to find the triangle's perimeter

00:37 We'll multiply by the reciprocal to isolate P

00:48 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$ΔDEF∼Δ\text{ABC}$

If the perimeter of $ΔABC$ is 64, then what is the perimeter of $ΔDEF$ ?

Step-by-step solution

To solve this problem, follow these steps:

Step 1: Find the ratio of similarity between the corresponding sides of the triangles $\triangle DEF$ and $\triangle ABC$ .
Step 2: Simplify the ratio of the sides.
Step 3: Apply the ratio to find the perimeter of $\triangle DEF$ .

Let's follow these steps:

Step 1: Given $EF = 18$ and $BC = 24$ are corresponding sides of similar triangles $\triangle DEF \sim \triangle ABC$ , we find the ratio:

$\frac{EF}{BC} = \frac{18}{24}$

Step 2: Simplify this ratio:

$\frac{18}{24} = \frac{3}{4}$

Step 3: The ratio of similarity $\frac{DE}{AB} = \frac{EF}{BC} = \frac{3}{4}$ means the perimeter of $\triangle DEF$ is $\frac{3}{4}$ of the perimeter of $\triangle ABC$ .

Given the perimeter of $\triangle ABC$ is 64, compute the perimeter of $\triangle DEF$ :

$\frac{3}{4} \times 64 = 48$

Therefore, the perimeter of $\triangle DEF$ is 48.

Final Answer

Key Points to Remember

Essential concepts to master this topic

Ratio Rule: Similar triangles have equal ratios for all corresponding sides
Technique: Find ratio using given sides: $\frac{18}{24} = \frac{3}{4}$
Check: Verify that $\frac{3}{4} \times 64 = 48$ produces reasonable triangle size ✓

Common Mistakes

Avoid these frequent errors

Confusing which triangle is larger and inverting the ratio
Don't use $\frac{24}{18} = \frac{4}{3}$ when DEF is the smaller triangle = perimeter of 85.3 which exceeds ABC's perimeter! This violates the similarity relationship shown in the diagram. Always identify which triangle is smaller by comparing the given corresponding sides first.

Practice Quiz

Test your knowledge with interactive questions

The two parallelograms above are similar. The ratio between their sides is 3:4.

What is the ratio between the the areas of the parallelograms?

FAQ

Everything you need to know about this question

How do I know which sides correspond to each other?

Look at the similarity statement $\triangle DEF \sim \triangle ABC$ . The order of vertices tells you: D↔A, E↔B, F↔C. So side EF corresponds to side BC, which are the sides given in the diagram.

Why can I use any pair of corresponding sides to find the ratio?

In similar triangles, all corresponding sides have the same ratio. Whether you use $\frac{DE}{AB}$ , $\frac{EF}{BC}$ , or $\frac{DF}{AC}$ , you'll get the same ratio!

Does the same ratio apply to perimeters?

Yes! If the side ratio is $\frac{3}{4}$ , then the perimeter ratio is also $\frac{3}{4}$ . This works because perimeter is just the sum of all sides, and each side has the same ratio.

What if I get a decimal ratio instead of a fraction?

That's fine! For example, if sides are 15 and 20, the ratio $\frac{15}{20} = 0.75$ . Just multiply the larger perimeter by 0.75 to get the smaller perimeter.

How can I tell which triangle is smaller from the diagram?

Compare the given side lengths: EF = 18 and BC = 24. Since 18 < 24, triangle DEF is smaller than triangle ABC. The smaller triangle will have the smaller perimeter.

What if I accidentally use the wrong ratio direction?

You'll get an impossible answer! If you use $\frac{4}{3} \times 64 \approx 85.3$ , the smaller triangle would have a larger perimeter than the bigger triangle, which makes no sense. Always check that your answer is reasonable!

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