Rectangle Diagonal Problem: Calculate Triangle Area with BO = 8.5 and AD = 8

Q: Why is BD equal to 2 times BO?

In any rectangle, the diagonals bisect each other at point O. This means O is exactly halfway along each diagonal, so BO = half of BD. Therefore, the full diagonal BD = 2 × BO.

Q: How do I know triangle ABD is a right triangle?

Triangle ABD has a right angle at A because ABCD is a rectangle. All corners of a rectangle are 90°, making ABD a right triangle with the diagonal BD as its hypotenuse.

Q: Can I use the diagonal formula instead of Pythagorean theorem?

Yes! For a rectangle with sides a and b, diagonal = $ \sqrt{a^2 + b^2} $. Since BD = 17 and AD = 8, you get: $ 17 = \sqrt{AB^2 + 8^2} $, leading to the same answer AB = 15.

Q: Why do I calculate the area using AD and AB instead of the diagonal?

Triangle area formula needs the base and height, which must be perpendicular. In right triangle ABD, sides AD and AB are perpendicular, so Area = $ \frac{1}{2} \times AD \times AB $.

Q: What if I calculated AB incorrectly?

Always verify using the Pythagorean theorem! If AB = 15, then $ 15^2 + 8^2 = 225 + 64 = 289 $, and $ \sqrt{289} = 17 $ ✓. This confirms your AB value is correct.

Rectangle Diagonal Theorem with Pythagorean Application

Below is the rectangle ABCD.

O is the intersection point of the diagonals of the rectangle.

AD = 8

BO = 8.5

Calculate the area of the triangle ABD.

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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

Below is the rectangle ABCD.

O is the intersection point of the diagonals of the rectangle.

AD = 8

BO = 8.5

Calculate the area of the triangle ABD.

Step-by-step solution

According to the given information, we can claim that:

$BD=2BO=8.5\times2=17$

Now let's look at triangle ABD to calculate side AB

$AB^2+AD^2=BD^2$

Let's input the known data:

$AB^2+8^2=17^2$

$AB^2=289-64=225$

We'll take the square root

$AB=15$

Now let's calculate the area of triangle ABD:

$\frac{15\times8}{2}=\frac{120}{2}=60$

Final Answer

Key Points to Remember

Essential concepts to master this topic

Diagonal Property: In a rectangle, diagonals bisect each other equally
Technique: Use $BD = 2 \times BO = 2 \times 8.5 = 17$
Check: Verify with Pythagorean: $15^2 + 8^2 = 225 + 64 = 289 = 17^2$ ✓

Common Mistakes

Avoid these frequent errors

Confusing BO with the full diagonal BD
Don't use BO = 8.5 as the full diagonal length = area of 30! The intersection point O divides each diagonal in half, so BO is only half of diagonal BD. Always remember BD = 2 × BO to get the complete diagonal length.

Practice Quiz

Test your knowledge with interactive questions

The points A and O are shown in the figure below.

Is it possible to draw a rectangle so that the side AO is its diagonal?

FAQ

Everything you need to know about this question

Why is BD equal to 2 times BO?

In any rectangle, the diagonals bisect each other at point O. This means O is exactly halfway along each diagonal, so BO = half of BD. Therefore, the full diagonal BD = 2 × BO.

How do I know triangle ABD is a right triangle?

Triangle ABD has a right angle at A because ABCD is a rectangle. All corners of a rectangle are 90°, making ABD a right triangle with the diagonal BD as its hypotenuse.

Can I use the diagonal formula instead of Pythagorean theorem?

Yes! For a rectangle with sides a and b, diagonal = $\sqrt{a^2 + b^2}$ . Since BD = 17 and AD = 8, you get: $17 = \sqrt{AB^2 + 8^2}$ , leading to the same answer AB = 15.

Why do I calculate the area using AD and AB instead of the diagonal?

Triangle area formula needs the base and height, which must be perpendicular. In right triangle ABD, sides AD and AB are perpendicular, so Area = $\frac{1}{2} \times AD \times AB$ .

What if I calculated AB incorrectly?

Always verify using the Pythagorean theorem! If AB = 15, then $15^2 + 8^2 = 225 + 64 = 289$ , and $\sqrt{289} = 17$ ✓. This confirms your AB value is correct.

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Rectangle Diagonal Problem: Calculate Triangle Area with BO = 8.5 and AD = 8

Rectangle Diagonal Theorem with Pythagorean Application

❤️ 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 is BD equal to 2 times BO?

How do I know triangle ABD is a right triangle?

Can I use the diagonal formula instead of Pythagorean theorem?

Why do I calculate the area using AD and AB instead of the diagonal?

What if I calculated AB incorrectly?

🌟 Unlock Your Math Potential

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