Calculate Rectangle Area: Finding ABGE with Given Measurements 5cm and 3cm

Q: Why can't I just use BC = 5 as the rectangle height?

Because BC is the hypotenuse of triangle BGC, not a side of the rectangle! The rectangle height is BG, which you must calculate using the Pythagorean theorem.

Q: What if I can't see the right triangle in the figure?

Look for the perpendicular lines (shown in gray). Point G is directly below B, making angle BGC a right angle. This creates the right triangle BGC with legs BG and GC.

Q: How do I remember the Pythagorean theorem formula?

Remember: $ a^2 + b^2 = c^2 $ where c is always the longest side (hypotenuse). In this problem, BC = 5 is the hypotenuse, so $ BG^2 + GC^2 = BC^2 $.

Q: Can I solve this problem without the Pythagorean theorem?

No! The Pythagorean theorem is essential here because you need to find the missing side BG of the right triangle. Without it, you can't determine the rectangle's height.

Pythagorean Theorem with Rectangle Area Calculation

The trapezoid ABCD and the rectangle ABGE are shown in the figure below.

Given in cm:

AB = 5

BC = 5

GC = 3

Calculate the area of the rectangle ABGE.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:09 Let's find the area of rectangle ABGE. Ready?

00:13 First, apply the Pythagorean theorem to triangle B, C, G.

00:17 Next, substitute the values you know, and solve for height B, G.

00:31 Good job! Now, isolate B, G by itself.

00:37 This B, G is the height, which is also a side of the rectangle.

00:42 To find the rectangle's area, multiply side four by side five.

00:49 And there you have it! That's the solution.

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

The trapezoid ABCD and the rectangle ABGE are shown in the figure below.

Given in cm:

AB = 5

BC = 5

GC = 3

Calculate the area of the rectangle ABGE.

Step-by-step solution

Let's calculate side BG using the Pythagorean theorem:

$BG^2+GC^2=BC^2$

We'll substitute the known data:

$BG^2+3^2=5^2$

$BG^2+9=25$

$BG^2=16$

$BG=\sqrt{16}=4$

Now we can calculate the area of rectangle ABGE since we have the length and width:

$5\times4=20$

Final Answer

Key Points to Remember

Essential concepts to master this topic

Right Triangle Rule: Use Pythagorean theorem when rectangle forms right triangle
Technique: $BG^2 + 3^2 = 5^2$ gives BG = 4
Check: Verify: $4^2 + 3^2 = 16 + 9 = 25 = 5^2$ ✓

Common Mistakes

Avoid these frequent errors

Using wrong formula for rectangle area
Don't assume the height equals a given side length without checking = wrong area calculation! Students often use AB × BC instead of finding the actual rectangle dimensions. Always identify which sides form the rectangle and use Pythagorean theorem when needed.

Practice Quiz

Test your knowledge with interactive questions

Look at the rectangle below.

Side DC has a length of 1.5 cm and side AD has a length of 9.5 cm.

What is the perimeter of the rectangle?

FAQ

Everything you need to know about this question

How do I know which sides form the rectangle ABGE?

Look at the figure carefully! Points A, B, G, E form the rectangle. Since AB = 5 is given and forms the top side, you need to find BG for the height using the right triangle BGC.

Why can't I just use BC = 5 as the rectangle height?

Because BC is the hypotenuse of triangle BGC, not a side of the rectangle! The rectangle height is BG, which you must calculate using the Pythagorean theorem.

What if I can't see the right triangle in the figure?

Look for the perpendicular lines (shown in gray). Point G is directly below B, making angle BGC a right angle. This creates the right triangle BGC with legs BG and GC.

How do I remember the Pythagorean theorem formula?

Remember: $a^2 + b^2 = c^2$ where c is always the longest side (hypotenuse). In this problem, BC = 5 is the hypotenuse, so $BG^2 + GC^2 = BC^2$ .

Can I solve this problem without the Pythagorean theorem?

No! The Pythagorean theorem is essential here because you need to find the missing side BG of the right triangle. Without it, you can't determine the rectangle's height.

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Calculate Rectangle Area: Finding ABGE with Given Measurements 5cm and 3cm

Pythagorean Theorem with Rectangle Area Calculation

❤️ 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

How do I know which sides form the rectangle ABGE?

Why can't I just use BC = 5 as the rectangle height?

What if I can't see the right triangle in the figure?

How do I remember the Pythagorean theorem formula?

Can I solve this problem without the Pythagorean theorem?

🌟 Unlock Your Math Potential

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