Similar Triangles: Calculate Perimeter Using 12-13-5 Triangle Measurements

Similar Triangles with Scale Factor Ratios

5.213125 The triangle above are similar.

What is the perimeter of the blue triangle?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:04 Let's find the perimeter of the triangle using the similarity ratio.
00:08 Remember, the perimeter of a triangle is the sum of all its sides.
00:13 Here, we have the known perimeter of one triangle.
00:19 The second triangle's perimeter is determined by the similarity ratio.
00:27 Let's plug in the values, and solve to find the new perimeter.
00:38 Next, we'll multiply by the reciprocal to find P.
00:49 And there you have it, that's our solution!

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

5.213125 The triangle above are similar.

What is the perimeter of the blue triangle?

2

Step-by-step solution

The perimeter of the left triangle: 13+12+5=25+5=30

Therefore, the perimeter of the right triangle divided by 30 is equal to 5.2 divided by 13:

x30=5.213 \frac{x}{30}=\frac{5.2}{13}

13x=156 13x=156

x=12 x=12

3

Final Answer

12

Key Points to Remember

Essential concepts to master this topic
  • Similar Triangles: Corresponding sides are proportional with same scale factor
  • Scale Factor: Find ratio by dividing corresponding sides: 5.213=0.4 \frac{5.2}{13} = 0.4
  • Check: All sides scale by same factor: 12×0.4=4.8, 13×0.4=5.2, 5×0.4=2.0 ✓

Common Mistakes

Avoid these frequent errors
  • Using wrong corresponding sides in ratio
    Don't match sides randomly like setting 5.2/12 = perimeter ratio! This creates incorrect proportions because 5.2 corresponds to 13, not 12. Always identify which sides correspond by position and angle, then use the correct pairs in your ratios.

Practice Quiz

Test your knowledge with interactive questions

If it is known that both triangles are equilateral, are they therefore similar?

FAQ

Everything you need to know about this question

How do I know which sides correspond to each other?

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Look at the position and angles! In similar triangles, corresponding sides are opposite the same angles. The side labeled 5.2 corresponds to the side labeled 13 because they're in the same relative position.

Why can't I just add the scale factor to each side?

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Similar triangles use multiplication, not addition! The scale factor is a multiplier. If the scale factor is 0.4, then each side of the smaller triangle equals the larger triangle's side × 0.4.

What if I get the scale factor upside down?

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Check if your answer makes sense! If the blue triangle looks smaller but your scale factor is greater than 1, you've flipped it. The scale factor from larger to smaller should be less than 1.

How do I find the perimeter once I know the scale factor?

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Two ways: Method 1: Find each side length first (12×0.4, 13×0.4, 5×0.4), then add them. Method 2: Find the large triangle's perimeter (30), then multiply by scale factor (30×0.4=12).

Can I use any pair of corresponding sides to find the scale factor?

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Yes! All corresponding sides have the same ratio in similar triangles. Whether you use 5.213 \frac{5.2}{13} or any other corresponding pair, you'll get the same scale factor of 0.4.

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