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

Question

Look at the following rectangle:

AAABBBCCCDDDEEEFFFGGGHHH105108

ΔEAG≅ΔFCH

Find the perimeter of rectangle EFCD.

Video Solution

Solution Steps

00:00 Find the perimeter EFCD
00:03 Equal sides according to triangle congruence
00:33 Place appropriate values according to the given data
00:45 The whole side (AB) equals the sum of its parts (AG+GB)
00:50 Place appropriate values and solve for AB
01:01 This is the length of 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 Use the Pythagorean theorem in triangle FHC to find FC
01:50 Place appropriate values and solve for FC
02:03 Isolate FC
02:21 This is the value of segment FC
02:26 Opposite sides are equal
02:29 The rectangle's perimeter equals the sum of its sides
02:41 Place appropriate values and solve for the perimeter
03:00 And this is the solution to the question

Step-by-Step Solution

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

AE=FC AE=FC

AG=CH=8 AG=CH=8

EG=FH=10 EG=FH=10

Now let's calculate side AB:

8+5=13 8+5=13

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

AB=CD=13 AB=CD=13

We can also claim that:

DH=138=5 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:

HC2+FC2=HF2 HC^2+FC^2=HF^2

Let's input the known data:

82+FC2=102 8^2+FC^2=10^2

64+FC2=100 64+FC^2=100

FC2=10064 FC^2=100-64

FC2=36 FC^2=36

Let's take the square root:

FC=6 FC=6

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

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

Answer

23+173 23+\sqrt{173}