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.

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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 EFCD

00:03 Equal sides according to the triangle congruence

00:33 Insert the relevant values into the formula according to the given data

00:45 The whole side (AB) equals the sum of its parts (AG+GB)

00:50 Insert the relevant values and proceed to solve for AB

01:01 This is the length of the side AB

01:04 Opposite sides in a rectangle are equal

01:13 The side segment (DH) equals the entire side (DC) minus segment (HC)

01:16 This is the size of DH

01:24 Equal sides

01:42 Apply the Pythagorean theorem to the triangle FHC in order to determine the value of FC

01:50 Insert the relevant values into the formula and proceed to solve for FC

02:03 Isolate FC

02:21 This is the value of the segment FC

02:26 Opposite sides are equal

02:29 The rectangle's perimeter equals the sum of its sides

02:41 Insert the relevant values into the formula and solve for the perimeter

03:00 This is the 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

Look at the rectangle ABCD below.

Side AB is 6 cm long and side BC is 4 cm long.

What is the area of the rectangle?

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 $6^2 + 8^2 = 10^2$ .

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