Similar Triangles in Rectangle ABCD: Calculate Perimeter Using 16, 8, and 10

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

Look at the order of vertices in the similarity statement! $ \triangle ADE \sim \triangle FCE $ means A↔F, D↔C, E↔E, so AD corresponds to FC and DE corresponds to CE.

Q: What if I get the ratio backwards?

Always check that your ratios are consistent! If $ \frac{AD}{FC} = \frac{DE}{CE} $, then $ \frac{12}{6} = \frac{16}{8} = 2 $. Both ratios should equal the same number.

Q: How do I find the rectangle's dimensions from the triangle?

The rectangle has width AD = 12 (from similar triangles) and length DC = DE + EC = 16 + 8 = 24. Then perimeter = 2(12 + 24) = 72.

Q: Can I solve this without similar triangles?

Similar triangles are the key insight here! The diagonal AC creates similar triangles, and this relationship lets you find the unknown side AD. Without recognizing the similarity, you can't determine the rectangle's dimensions.

Similar Triangles with Pythagorean Theorem

$ΔADE∼Δ\text{FCE}$

Calculate the perimeter of the given rectangle ABCD.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Determine the perimeter of the rectangle ABCD

00:03 Apply the Pythagorean theorem to the triangle EFC

00:08 Substitute the relevant values into the equation and solve for FC

00:14 Isolate FC

00:20 This is the length of FC

00:26 The triangles are similar according to the given information, proceed to determine the similarity ratio

00:33 Divide a side in one triangle by its corresponding side in the second triangle

00:46 Substitute the relevant side values into the formula in order to determine the ratio

00:58 Multiply by 6 to obtain the length of AD

01:03 This is the length of side AD, now we can calculate the rectangle's perimeter

01:07 Opposite sides are equal in a rectangle

01:14 The side equals the sum of its parts

01:18 The perimeter of the rectangle equals the sum of its sides

01:22 Therefore, substitute the side values into the formula and add them together in order to determine the perimeter

01:26 This is the solution

Step-by-step written solution

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Understand the problem

$ΔADE∼Δ\text{FCE}$

Calculate the perimeter of the given rectangle ABCD.

Step-by-step solution

Let's begin by observing triangle FCE and calculate side FC using the Pythagorean theorem:

$EC^2+FC^2=EF^2$

Let's begin by substituting all the known values into the formula:

$8^2+FC^2=10^2$

$64+FC^2=100$

$FC^2=100-64$

$FC^2=36$

Let's take the square root:

$FC=6$

Since we know that the triangles overlap:

$\frac{AD}{FC}=\frac{DE}{CE}=\frac{AE}{FE}$

Let's again substitute the known values into the formula:

$\frac{AD}{6}=\frac{16}{8}$

$AD=2\times6=12$

Finally let's calculate side CD:

$16+8=24$

Since in a rectangle each pair of opposite sides are equal, we can calculate the perimeter of rectangle ABCD as follows:

$12+24+12+24=24+48=72$

Final Answer

Key Points to Remember

Essential concepts to master this topic

Similar Triangles: Corresponding sides have equal ratios in similar triangles
Pythagorean Theorem: Find FC using $8^2 + FC^2 = 10^2$ , so FC = 6
Check: Verify ratio AD/FC = DE/CE gives 12/6 = 16/8 = 2 ✓

Common Mistakes

Avoid these frequent errors

Not using Pythagorean theorem to find missing side first
Don't try to set up similar triangle ratios without finding FC first = incomplete information! You need all three sides of triangle FCE to establish the correct similarity ratio. Always use Pythagorean theorem to find FC = 6 before setting up proportions.

Practice Quiz

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

FAQ

Everything you need to know about this question

Why do I need to use the Pythagorean theorem first?

You need to find the missing side FC before you can use similar triangles! Triangle FCE has sides EC = 8 and EF = 10, so use $FC^2 + 8^2 = 10^2$ to get FC = 6.

How do I know which sides correspond in similar triangles?

Look at the order of vertices in the similarity statement! $\triangle ADE \sim \triangle FCE$ means A↔F, D↔C, E↔E, so AD corresponds to FC and DE corresponds to CE.

What if I get the ratio backwards?

Always check that your ratios are consistent! If $\frac{AD}{FC} = \frac{DE}{CE}$ , then $\frac{12}{6} = \frac{16}{8} = 2$ . Both ratios should equal the same number.

How do I find the rectangle's dimensions from the triangle?

The rectangle has width AD = 12 (from similar triangles) and length DC = DE + EC = 16 + 8 = 24. Then perimeter = 2(12 + 24) = 72.

Can I solve this without similar triangles?

Similar triangles are the key insight here! The diagonal AC creates similar triangles, and this relationship lets you find the unknown side AD. Without recognizing the similarity, you can't determine the rectangle's dimensions.

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Similar Triangles in Rectangle ABCD: Calculate Perimeter Using 16, 8, and 10

Similar Triangles with Pythagorean Theorem

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why do I need to use the Pythagorean theorem first?

How do I know which sides correspond in similar triangles?

What if I get the ratio backwards?

How do I find the rectangle's dimensions from the triangle?

Can I solve this without similar triangles?

🌟 Unlock Your Math Potential

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