Similar Triangles and Rectangle EFGD: Calculate Perimeter Using 3, 6, and 8 Units

Q: How do I know which sides correspond in similar triangles?

Look at the order of vertices in the similarity statement! In $ \triangle BCD \sim \triangle FED $, B corresponds to F, C corresponds to E, and D corresponds to D.

Q: Why is the rectangle EFGD and not EFGC?

From the diagram, point G is directly below F, making EFGD a rectangle. The sides EF and GD are horizontal, while ED and FG are vertical.

Q: What if I get a different proportion setup?

As long as you match corresponding sides correctly, different valid setups like $ \frac{3}{8} = \frac{EF}{6} $ will give the same answer: EF = 2.25.

Q: How do I calculate the rectangle perimeter once I know the sides?

Use the formula: Perimeter = 2(length + width). Here: P = 2(EF + ED) = 2(2.25 + 3) = 2(5.25) = 10.5

Similar Triangles with Proportional Rectangle Dimensions

ΔBCD∼ΔFED

Calculate the perimeter of the rectangle EFGD.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the perimeter of rectangle EFGD

00:05 Similar triangles according to the given data

00:10 Find the similarity ratio

00:32 Substitute appropriate values

00:47 Isolate FE

01:13 This is the length of FE

01:20 Opposite sides are equal in a rectangle

01:34 Side length according to the given data

01:39 The perimeter of the rectangle equals the sum of its sides

01:50 Substitute appropriate values and solve to find the perimeter

02:09 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

ΔBCD∼ΔFED

Calculate the perimeter of the rectangle EFGD.

Step-by-step solution

To solve for the perimeter of rectangle $EFGD$ , we follow these steps:

Step 1: Use the similarity of the triangles $\triangle BCD \sim \triangle FED$ . This implies the sides are proportional:

\frac{CD}{ED} = \frac{BC}{EF}

Substitute the known lengths $CD = 8$ and $ED = 3$ :

\frac{8}{3} = \frac{6}{EF}

Solving for $EF$ , cross-multiply:

8 \times EF = 3 \times 6

EF = \frac{18}{8} = 2.25

Step 2: The length and width of rectangle $EFGD$ are $2.25$ (as calculated from $EF$ ) and $3$ (equal to $ED$ ).

Step 3: Calculate the perimeter $P$ of $EFGD$ :

P = 2 \times (EF + ED) = 2 \times (2.25 + 3) = 2 \times 5.25

P = 10.5

The perimeter of rectangle $EFGD$ is $10.5$ .

Final Answer

10.5

Key Points to Remember

Essential concepts to master this topic

Similarity Rule: Corresponding sides are proportional in similar triangles
Technique: Set up proportion $\frac{8}{3} = \frac{6}{EF}$ to find EF = 2.25
Check: Rectangle perimeter is 2(length + width) = 2(2.25 + 3) = 10.5 ✓

Common Mistakes

Avoid these frequent errors

Using wrong corresponding sides in proportion
Don't match CD with EF instead of ED = wrong ratio and wrong answer! This mixes up which sides correspond between similar triangles. Always identify corresponding sides carefully: CD corresponds to ED, and BC corresponds to EF.

Practice Quiz

Test your knowledge with interactive questions

Look at the rectangle below.

Side DC has a length of 1.5 cm and side AD has a length of 9.5 cm.

What is the perimeter of the rectangle?

FAQ

Everything you need to know about this question

How do I know which sides correspond in similar triangles?

Look at the order of vertices in the similarity statement! In $\triangle BCD \sim \triangle FED$ , B corresponds to F, C corresponds to E, and D corresponds to D.

Why is the rectangle EFGD and not EFGC?

From the diagram, point G is directly below F, making EFGD a rectangle. The sides EF and GD are horizontal, while ED and FG are vertical.

Can I use the diagonal AC to solve this problem?

No, the diagonal AC doesn't give us the proportional relationship we need. Focus on the corresponding sides of the similar triangles BCD and FED.

What if I get a different proportion setup?

As long as you match corresponding sides correctly, different valid setups like $\frac{3}{8} = \frac{EF}{6}$ will give the same answer: EF = 2.25.

How do I calculate the rectangle perimeter once I know the sides?

Use the formula: Perimeter = 2(length + width). Here: P = 2(EF + ED) = 2(2.25 + 3) = 2(5.25) = 10.5

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