Calculate Triangle Area: Finding BCD with Side Lengths 12 and 13

Q: Why do I need to find AD first when calculating triangle BCD area?

Triangle BCD needs both a base and height to calculate area. We know BC = 12 from the rectangle, but we need CD = AD = 5. Using the Pythagorean theorem on right triangle ABC gives us this missing measurement!

Q: What if I calculated AD wrong using the Pythagorean theorem?

Double-check your calculation: $ 13^2 = 169 $ and $ 12^2 = 144 $, so $ AD^2 = 169 - 144 = 25 $, giving $ AD = 5 $. If you get a different value, your final triangle area will be wrong too!

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

Great observation! Triangle BCD has the same base and height as the rectangle (12 × 5), so its area is $ \frac{1}{2} \times 12 \times 5 = 30 $. The rectangle area is $ 12 \times 5 = 60 $. This always happens when a triangle uses two adjacent sides of a rectangle!

Rectangle Triangle Area with Pythagorean Theorem

Look at the following rectangle:

AB = 12

AC = 13

Calculate the area of the triangle BCD.

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Step-by-step video solution

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Step-by-step written solution

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Understand the problem

Look at the following rectangle:

AB = 12

AC = 13

Calculate the area of the triangle BCD.

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Use the Pythagorean Theorem to calculate the length of $AD$ .
Step 2: Calculate the area of triangle $\triangle BCD$ .

Step 1: Given $AB = 12$ and $AC = 13$ , we use the Pythagorean Theorem to find $AD$ .

AC^2 = AB^2 + AD^2 \implies 13^2 = 12^2 + AD^2

169 = 144 + AD^2 \implies AD^2 = 25 \implies AD = 5

Step 2: Knowing the sides $AD = 5$ (height of the rectangle) and $AB = 12$ (base of the rectangle), triangle $\triangle BCD$ will have the base $BC = 12$ and the height $BD = 5$ .

The area of triangle $\triangle BCD$ is:

\text{Area}_{\triangle BCD} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 12 \times 5 = 30

Therefore, the area of triangle $\triangle BCD$ is 30.

Final Answer

Key Points to Remember

Essential concepts to master this topic

Rectangle Properties: Opposite sides equal, all angles are 90 degrees
Pythagorean Theorem: $AC^2 = AB^2 + AD^2$ gives $13^2 = 12^2 + AD^2$
Triangle Area Check: Base × height ÷ 2 = 12 × 5 ÷ 2 = 30 ✓

Common Mistakes

Avoid these frequent errors

Using the diagonal as base or height for triangle area
Don't use AC = 13 as the base or height for triangle BCD = wrong area calculation! The diagonal connects opposite corners and isn't a side of the triangle. Always use the rectangle's actual sides (BC = 12 and CD = 5) for the triangle area formula.

Practice Quiz

Test your knowledge with interactive questions

Complete the sentence:

To find the area of a right triangle, one must multiply ________________ by each other and divide by 2.

FAQ

Everything you need to know about this question

Why do I need to find AD first when calculating triangle BCD area?

Triangle BCD needs both a base and height to calculate area. We know BC = 12 from the rectangle, but we need CD = AD = 5. Using the Pythagorean theorem on right triangle ABC gives us this missing measurement!

How do I know which sides of the rectangle to use for the triangle?

Triangle BCD uses three vertices of the rectangle: B, C, and D. So its sides are BC (which equals AB = 12), CD (which equals AD = 5), and diagonal BD. For area, use the two perpendicular sides BC and CD.

Can I calculate the area using BD as the base instead?

Yes, but it's more complicated! You'd need to find BD using $BD^2 = BC^2 + CD^2 = 12^2 + 5^2 = 169$ , so BD = 13. Then you'd need the perpendicular height to BD, which requires more advanced geometry.

What if I calculated AD wrong using the Pythagorean theorem?

Double-check your calculation: $13^2 = 169$ and $12^2 = 144$ , so $AD^2 = 169 - 144 = 25$ , giving $AD = 5$ . If you get a different value, your final triangle area will be wrong too!

Why is the triangle area exactly half the rectangle area?

Great observation! Triangle BCD has the same base and height as the rectangle (12 × 5), so its area is $\frac{1}{2} \times 12 \times 5 = 30$ . The rectangle area is $12 \times 5 = 60$ . This always happens when a triangle uses two adjacent sides of a rectangle!

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Calculate Triangle Area: Finding BCD with Side Lengths 12 and 13

Rectangle Triangle Area with Pythagorean Theorem

❤️ 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 I need to find AD first when calculating triangle BCD area?

How do I know which sides of the rectangle to use for the triangle?

Can I calculate the area using BD as the base instead?

What if I calculated AD wrong using the Pythagorean theorem?

Why is the triangle area exactly half the rectangle area?

🌟 Unlock Your Math Potential

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