Right Triangle Side Length: Find BC When Area = 32 and Height = 8

Q: How do I know which side is the base and which is the height?

In a right triangle, any of the two perpendicular sides can be the base or height! The key is that they must be the sides that form the right angle, not the hypotenuse.

Q: Can I use the hypotenuse in the area formula?

No! The area formula $ \frac{1}{2} \times \text{base} \times \text{height} $ only works with perpendicular sides. The hypotenuse is never perpendicular to itself.

Q: What if I get a decimal or fraction for the side length?

That's completely normal! Side lengths can be any positive number, including decimals and fractions. Just make sure your calculation is accurate.

Q: Why do we multiply by 1/2 in the area formula?

A right triangle is half of a rectangle! If you draw a rectangle with the same base and height, the triangle takes up exactly half the space.

Q: How can I double-check my work?

Substitute your answer back into the area formulaMake sure $ \frac{1}{2} \times 8 \times 8 = 32 $Check that your answer is positive (side lengths can't be negative!)

Right Triangle Area with Known Height

ABC is a right triangle with an area of 32.

Calculate the length of side BC.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Determine the value of BC

00:02 Apply the formula to calculating the triangle's area

00:06 (Height(AB) x base(BC)) divided by 2

00:12 Substitute in the relevant values and proceed to calculate to determine BC

00:20 Divide 8 by 2 to obtain 4

00:24 Isolate BC

00:30 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

ABC is a right triangle with an area of 32.

Calculate the length of side BC.

Step-by-step solution

To solve this problem, we need to calculate the length of side $BC$ in triangle $\triangle ABC$ given that the area is 32 and side $AB = 8$ .

We start by using the area formula for a right triangle:

\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}

In this context, the base $AB$ is 8, and the height $BC$ is the unknown we need to find. Thus, we have:

\frac{1}{2} \times 8 \times BC = 32

We can simplify this equation:

4 \times BC = 32

Now, solve for $BC$ by dividing both sides of the equation by 4:

BC = \frac{32}{4} = 8

Therefore, the length of side $BC$ is $\mathbf{8}$ .

Thus, the solution to the problem is $BC = 8$ .

Final Answer

Key Points to Remember

Essential concepts to master this topic

Formula: Area = $\frac{1}{2} \times \text{base} \times \text{height}$ for right triangles
Technique: Substitute known values: $\frac{1}{2} \times 8 \times BC = 32$
Check: Verify by calculating area: $\frac{1}{2} \times 8 \times 8 = 32$ ✓

Common Mistakes

Avoid these frequent errors

Confusing which sides are base and height
Don't assume the hypotenuse can be used as base or height = wrong formula application! In a right triangle, only the two perpendicular sides (legs) can serve as base and height. Always identify the right angle first and use the two sides that form it.

Practice Quiz

Test your knowledge with interactive questions

Angle A is equal to 30°.
Angle B is equal to 60°.
Angle C is equal to 90°.

Can these angles form a triangle?

FAQ

Everything you need to know about this question

How do I know which side is the base and which is the height?

In a right triangle, any of the two perpendicular sides can be the base or height! The key is that they must be the sides that form the right angle, not the hypotenuse.

Can I use the hypotenuse in the area formula?

No! The area formula $\frac{1}{2} \times \text{base} \times \text{height}$ only works with perpendicular sides. The hypotenuse is never perpendicular to itself.

What if I get a decimal or fraction for the side length?

That's completely normal! Side lengths can be any positive number, including decimals and fractions. Just make sure your calculation is accurate.

Why do we multiply by 1/2 in the area formula?

A right triangle is half of a rectangle! If you draw a rectangle with the same base and height, the triangle takes up exactly half the space.

How can I double-check my work?

Substitute your answer back into the area formula
Make sure $\frac{1}{2} \times 8 \times 8 = 32$
Check that your answer is positive (side lengths can't be negative!)

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