Rectangle Perimeter with Congruent Triangles: Finding Total Length Using 10, 5, and 8

Q: How do I know which sides are equal in congruent triangles?

Corresponding sides in congruent triangles are equal. Since ΔEAG ≅ ΔFCH, we have: EA = FC, AG = CH = 8, and EG = FH = 10.

Q: Why can't I just add up the visible measurements?

You need to find the unknown side lengths first! The rectangle sides aren't all labeled, so use the Pythagorean theorem and congruent triangle properties to calculate missing lengths.

Q: How do I find FC if it's not directly labeled?

Use triangle FCH with the Pythagorean theorem: $ FC^2 + 8^2 = 10^2 $, so $ FC^2 = 100 - 64 = 36 $, giving FC = 6.

Congruent Triangles with Rectangle Perimeter

Look at the following rectangle:

ΔEAG≅ΔFCH

Find the perimeter of rectangle EFCD.

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:09 Let's find the perimeter of rectangle EFCD.

00:13 Remember, congruent triangles have equal sides.

00:42 Now, let's plug in the values from our data into the formula.

00:54 Side A B is equal to A G plus G B.

00:59 Next, use these values to find the length of side A B.

01:10 This is the length of side A B.

01:13 Remember, opposite sides in a rectangle are equal.

01:22 For D H, subtract segment H C from D C.

01:27 And here's the length of D H.

01:33 These are your equal sides.

01:51 To find F C, use the Pythagorean theorem on triangle F H C.

01:59 Insert the known values and solve for F C.

02:12 Isolate F C in your equation.

02:30 And this is the value of segment F C.

02:35 Opposite sides are equal here too.

02:38 The rectangle's perimeter is the sum of all its sides.

02:50 Put the values into the formula and find the perimeter.

03:09 Great job! This is your solution.

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Look at the following rectangle:

ΔEAG≅ΔFCH

Find the perimeter of rectangle EFCD.

Step-by-step solution

Since the triangles are equal to each other, we can claim that:

$AE=FC$

$AG=CH=8$

$EG=FH=10$

Now let's calculate side AB:

$8+5=13$

Since in a rectangle each pair of opposite sides are equal to each other:

$AB=CD=13$

We can also claim that:

$DH=13-8=5$

Side EF is also equal in length to sides AB and CD which are equal to 13

Now let's calculate side FC using the Pythagorean theorem in triangle FCH:

$HC^2+FC^2=HF^2$

Let's input the known data:

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

$64+FC^2=100$

$FC^2=100-64$

$FC^2=36$

Let's take the square root:

$FC=6$

Now we can calculate the perimeter of rectangle EFCD by adding all sides together:

$13+6+13+6=26+12=38$

Final Answer

$23+\sqrt{173}$

Key Points to Remember

Essential concepts to master this topic

Congruence: Equal triangles have corresponding sides and angles equal
Pythagorean Theorem: Use $a^2 + b^2 = c^2$ to find FC = 6
Verification: Check that $8^2 + 6^2 = 10^2$ gives 100 = 100 ✓

Common Mistakes

Avoid these frequent errors

Forgetting to use congruent triangle properties
Don't calculate side lengths without using ΔEAG ≅ ΔFCH = wrong dimensions! This ignores the given congruence and leads to incorrect perimeter calculations. Always identify corresponding equal sides from congruent triangles first.

Practice Quiz

Test your knowledge with interactive questions

Calculate the perimeter of the rectangle below.

FAQ

Everything you need to know about this question

How do I know which sides are equal in congruent triangles?

Corresponding sides in congruent triangles are equal. Since ΔEAG ≅ ΔFCH, we have: EA = FC, AG = CH = 8, and EG = FH = 10.

Why can't I just add up the visible measurements?

You need to find the unknown side lengths first! The rectangle sides aren't all labeled, so use the Pythagorean theorem and congruent triangle properties to calculate missing lengths.

How do I find FC if it's not directly labeled?

Use triangle FCH with the Pythagorean theorem: $FC^2 + 8^2 = 10^2$ , so $FC^2 = 100 - 64 = 36$ , giving FC = 6.

What's the difference between this and a regular rectangle problem?

Regular rectangles give you direct side lengths. Here, you must use congruent triangles to find the relationships between sides, then apply geometry theorems to calculate unknown lengths.

How do I verify my perimeter is correct?

Check that your calculated sides make sense: EF should equal the rectangle width (13), and EC should be the height. Also verify FC = 6 satisfies $6^2 + 8^2 = 10^2$ .

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Rectangles questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime