Find Rectangle Area: Using Triangle Area 8X and Side Length X+4

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

Triangle ABC is exactly half of rectangle ABCD since the diagonal divides it into two equal triangles. Using the triangle area 8X helps us find the variable X first!

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

In a rectangle, adjacent sides are perpendicular, so any two connecting sides work as base and height. Here, BC = X and AB = X+4 are perfect choices.

Q: What if I get a negative value for X?

Since X represents a length measurement, it must be positive! If you get negative, check your algebra steps - you might have made an error in solving the equation.

Q: Why multiply both sides by 2 instead of dividing by 1/2?

Both methods work! Multiplying by 2 often feels easier: $ 8x = \frac{(x+4)x}{2} $ becomes $ 16x = (x+4)x $. Choose whichever feels more comfortable to you.

Triangle Area Formula with Variable Dimensions

Given the rectangle ABCD

Given BC=X and the side AB is larger by 4 cm than the side BC.

The area of the triangle ABC is 8X cm².

What is the area of the rectangle?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Calculate the area of the rectangle ABCD

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

00:08 (base(X+4) x height(X)) divided by 2

00:14 Substitute in the relevant values and proceed to solve for X

00:20 Multiply by 2 to eliminate the fraction

00:27 Reduce X from both sides of the equation

00:37 This is the length of side BC

00:47 The area of rectangle ABCD equals twice the area of triangle ABC

00:57 This is true given that triangles ABC and ACD are congruent (S.S.S.)

01:01 Substitute in the relevant values and solve for the area

01:16 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Given the rectangle ABCD

Given BC=X and the side AB is larger by 4 cm than the side BC.

The area of the triangle ABC is 8X cm².

What is the area of the rectangle?

Step-by-step solution

Let's calculate the area of triangle ABC:

$8x=\frac{(x+4)x}{2}$

Multiply by 2:

$16x=(x+4)x$

Divide by x:

$16=x+4$

Let's move 4 to the left side and change the sign accordingly:

$16-4=x$

$12=x$

Now let's calculate the area of the rectangle, multiply the length and width where BC equals 12 and AB equals 16:

$16\times12=192$

Final Answer

192

Key Points to Remember

Essential concepts to master this topic

Triangle Area Formula: Area = $\frac{1}{2} \times \text{base} \times \text{height}$ for right triangles
Set up equation: $8x = \frac{(x+4) \times x}{2}$ then solve for x
Verification: Check x=12: Triangle area = $\frac{16 \times 12}{2} = 96$ , and $8 \times 12 = 96$ ✓

Common Mistakes

Avoid these frequent errors

Confusing triangle area with rectangle area
Don't use $8x = (x+4) \times x$ directly = wrong setup! This treats the triangle area as if it equals length × width. Always remember triangle area needs the $\frac{1}{2}$ factor in the formula.

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

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

Triangle ABC is exactly half of rectangle ABCD since the diagonal divides it into two equal triangles. Using the triangle area 8X helps us find the variable X first!

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

In a rectangle, adjacent sides are perpendicular, so any two connecting sides work as base and height. Here, BC = X and AB = X+4 are perfect choices.

What if I get a negative value for X?

Since X represents a length measurement, it must be positive! If you get negative, check your algebra steps - you might have made an error in solving the equation.

Why multiply both sides by 2 instead of dividing by 1/2?

Both methods work! Multiplying by 2 often feels easier: $8x = \frac{(x+4)x}{2}$ becomes $16x = (x+4)x$ . Choose whichever feels more comfortable to you.

How do I check my final rectangle area?

Substitute X=12: Length = 12+4 = 16, Width = 12. Rectangle area = 16 × 12 = 192. Also verify the triangle area matches: $\frac{16 \times 12}{2} = 96 = 8 \times 12$ ✓

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Find Rectangle Area: Using Triangle Area 8X and Side Length X+4

Triangle Area Formula with Variable Dimensions

❤️ 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

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

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

What if I get a negative value for X?

Why multiply both sides by 2 instead of dividing by 1/2?

How do I check my final rectangle area?

🌟 Unlock Your Math Potential

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