Rectangle Perimeter with Congruent Triangles: Using 5 and 3 Units

Q: What does ΔDEO ≅ ΔBFO actually tell me?

The symbol ≅ means congruent, so these triangles are identical in shape and size. This means corresponding sides are equal: DE = BF, EO = FO, and DO = BO.

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

Look at the order of vertices in the congruence statement. ΔDEO ≅ ΔBFO means D↔B, E↔F, and O↔O. So DE corresponds to BF, EO to FO, and DO to BO.

Q: Why do I use the Pythagorean theorem here?

Triangle BFO is a right triangle (you can see the right angle at F in the diagram). Since we know OF = 3 and BO = 5, we can find BF using $ 3^2 + BF^2 = 5^2 $.

Q: How does knowing BF = 4 help me find the rectangle's perimeter?

Since the triangles are congruent, DE = BF = 4. Looking at the rectangle, BC = BF + FC = 4 + 4 = 8. In rectangles, opposite sides are equal, so AD = BC = 8 and AB = DC = 6.

Q: What if I can't see the right angles in the diagram?

Look for perpendicular lines or check if the numbers fit the Pythagorean theorem. Here, $ 3^2 + 4^2 = 9 + 16 = 25 = 5^2 $, confirming we have right triangles.

Rectangle Perimeters with Congruent Triangles

Look at the following rectangle:

ΔDEO ≅ ΔBFO

Calculate the perimeter of the rectangle ABCD.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:04 Let's calculate the perimeter of the rectangle.

00:10 Equal sides can be found using triangle congruence.

00:16 Remember, a whole side equals the sum of its parts.

00:25 Now, substitute the given values and calculate to find the length of E F.

00:33 In a rectangle, opposite sides are equal.

00:43 Use the Pythagorean theorem in triangle B O F to find the length of B F.

00:54 Substitute the appropriate values and solve to find B F.

01:06 Let's isolate B F.

01:20 And there you have it, the value of segment B F.

01:30 Next, use the congruence ratio to find the length of E D.

01:45 Sides are equal because the whole side is equal and the segments are equal.

01:50 Again, a whole side equals the sum of its parts.

01:55 Remember, opposite sides are equal in a rectangle.

02:00 The perimeter is simply the sum of all sides.

02:04 Substitute the values and solve to find the rectangle's perimeter.

02:26 And that's how we solve this problem.

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Look at the following rectangle:

ΔDEO ≅ ΔBFO

Calculate the perimeter of the rectangle ABCD.

Step-by-step solution

Based on the given data, we can claim that:

$OF=OE=3$

$EF=6$

$AB=EF=DC=6$

We'll find side BF using the Pythagorean theorem in triangle BFO:

$OF^2+BF^2=BO^2$

Let's substitute the known values into the formula:

$3^2+BF^2=5^2$

$9+BF^2=25$

$BF^2=25-9$

$BF^2=16$

Let's take the square root:

$BF=4$

Since the triangles overlap:

$BF=DE=4=FC$

From this, we can calculate side BC:

$BC=4+4=8$

Since in a rectangle, each pair of opposite sides are equal to each other, we can claim that AD also equals 8

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

$6+8+6+8=12+16=28$

Final Answer

Key Points to Remember

Essential concepts to master this topic

Congruence: Congruent triangles have identical corresponding sides and angles
Pythagorean Theorem: Use $a^2 + b^2 = c^2$ to find BF = 4
Check: Verify perimeter: 6 + 8 + 6 + 8 = 28 units ✓

Common Mistakes

Avoid these frequent errors

Not recognizing that congruent triangles provide equal corresponding sides
Don't assume triangle sides are different just because they're in different positions = missing key measurements! Triangle congruence means all corresponding parts are equal. Always identify which sides correspond between congruent triangles to find unknown measurements.

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

What does ΔDEO ≅ ΔBFO actually tell me?

The symbol ≅ means congruent, so these triangles are identical in shape and size. This means corresponding sides are equal: DE = BF, EO = FO, and DO = BO.

How do I know which sides correspond in congruent triangles?

Look at the order of vertices in the congruence statement. ΔDEO ≅ ΔBFO means D↔B, E↔F, and O↔O. So DE corresponds to BF, EO to FO, and DO to BO.

Why do I use the Pythagorean theorem here?

Triangle BFO is a right triangle (you can see the right angle at F in the diagram). Since we know OF = 3 and BO = 5, we can find BF using $3^2 + BF^2 = 5^2$ .

How does knowing BF = 4 help me find the rectangle's perimeter?

Since the triangles are congruent, DE = BF = 4. Looking at the rectangle, BC = BF + FC = 4 + 4 = 8. In rectangles, opposite sides are equal, so AD = BC = 8 and AB = DC = 6.

What if I can't see the right angles in the diagram?

Look for perpendicular lines or check if the numbers fit the Pythagorean theorem. Here, $3^2 + 4^2 = 9 + 16 = 25 = 5^2$ , confirming we have right triangles.

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