Rectangle Perimeter Problem: Using Triangle Area of 9 to Find Solution

Q: Why do we use triangle ABD instead of the whole rectangle?

The diagonal AD creates triangle ABD with a right angle at A. This gives us a triangle with clearly defined base (AB = 6) and height (AD = x+2), making the area formula easy to apply.

Q: How do I know which sides are the base and height?

In rectangle ABCD, sides AB and AD are perpendicular (meet at 90°). So either can be base or height - just stay consistent! Here we used AB = 6 as base and AD = x+2 as height.

Q: Why is the triangle area exactly half the rectangle area?

The diagonal divides the rectangle into two identical triangles. Since triangle ABD has area 9, the full rectangle ABCD has area = 2 × 9 = 18.

Triangle Area with Rectangle Perimeter

Look at the following rectangle:

Given that the area of the triangle ABD is 9, what is the perimeter of the rectangle ABCD?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Calculate the perimeter of the rectangle

00:03 Apply the formula for calculating the area of a triangle

00:07 (height X side) divided by 2

00:27 Divide 6 by 2

00:34 Open the parentheses, multiply by each factor

00:44 Isolate X

00:53 This is the value of X, substitute in the relevant value to determine the side length

01:09 Opposite sides are equal in a rectangle

01:23 The perimeter of the rectangle equals the sum of its sides

01:30 Substitute in the relevant values and proceed to solve for the perimeter

01:45 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:

Given that the area of the triangle ABD is 9, what is the perimeter of the rectangle ABCD?

Step-by-step solution

Area of triangle ADB:

$\frac{AD\times AB}{2}$

Let's list the known data:

$9=\frac{x+2\times6}{2}$

$9=(x+2)\times3$

$9=3x+6$

$9-6=3x$

$3=3x$

$1=x$

Side AD equals:

$1+2=3$

Since in a rectangle, each pair of opposite sides are equal, we can state that:

$AD=BC=3$

$AB=CD=6$

Now we can calculate the perimeter of the rectangle:

$3+6+3+6=6+12=18$

Final Answer

Key Points to Remember

Essential concepts to master this topic

Triangle Area Formula: Area = $\frac{base \times height}{2}$ for right triangles
Setup Equation: $9 = \frac{6 \times (x+2)}{2}$ gives us $9 = 3(x+2)$
Verify Solution: When x=1, triangle area = $\frac{6 \times 3}{2} = 9$ ✓

Common Mistakes

Avoid these frequent errors

Using wrong triangle area formula
Don't use Area = base × height without dividing by 2 = double the correct area! This gives you 18 instead of 9, leading to wrong x-value. Always remember triangle area needs the ÷2 factor.

Practice Quiz

Test your knowledge with interactive questions

Calculate the perimeter of the rectangle below.

FAQ

Everything you need to know about this question

Why do we use triangle ABD instead of the whole rectangle?

The diagonal AD creates triangle ABD with a right angle at A. This gives us a triangle with clearly defined base (AB = 6) and height (AD = x+2), making the area formula easy to apply.

How do I know which sides are the base and height?

In rectangle ABCD, sides AB and AD are perpendicular (meet at 90°). So either can be base or height - just stay consistent! Here we used AB = 6 as base and AD = x+2 as height.

Why is the triangle area exactly half the rectangle area?

The diagonal divides the rectangle into two identical triangles. Since triangle ABD has area 9, the full rectangle ABCD has area = 2 × 9 = 18.

What if I solve for the wrong variable?

Always identify what the problem asks for! Here we found x = 1, but the question wants the perimeter. Don't forget to calculate 2(AB + AD) = 2(6 + 3) = 18.

How can I check my perimeter answer?

Verify that your rectangle sides make sense: AB = CD = 6 and AD = BC = 3. Then perimeter = 6 + 3 + 6 + 3 = 18. Also check that triangle area = $\frac{6 \times 3}{2} = 9$ ✓

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