Right Triangle Side Length: Find BC When Area = 40 and Height = 10

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

In a right triangle, the base and height are always the two sides that meet at the right angle. From the diagram, AB (length 10) and BC are perpendicular, so either can be base or height.

Q: Why do we use the formula Area = (1/2) × base × height?

This formula works because a triangle is exactly half of a rectangle. If you draw a rectangle with the same base and height, the triangle takes up half the space!

Q: What if I get a different answer when I switch base and height?

You should get the same answer either way! If AB = 10 is the base and BC is the height, or vice versa, the calculation $ \frac{1}{2} \times 10 \times BC = 40 $ gives BC = 8.

Q: Can I use this method for any triangle?

This specific method works for right triangles where you know the area and one leg. For other triangles, you might need different formulas or additional information like angles.

Q: How do I check if my answer makes sense?

Ask yourself: Does this create a reasonable triangle? With sides 8 and 10, and area 40, substitute back: $ \frac{1}{2} \times 8 \times 10 = 40 $ ✓

Triangle Area with Given Measurements

ABC is a right triangle with an area of 40.

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 Calculate the area of triangle ABC

00:03 Apply the formula for 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

00:22 Multiply by 2 to avoid fractions

00:28 Divide by 10 to isolate BC

00:37 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 40.

Calculate the length of side BC.

Step-by-step solution

The problem provides the area of a right triangle $ABC$ , which is 40, and tells us that $AB = 10$ , one of the legs. We need to find the base $BC$ of the triangle.

To find $BC$ , we use the formula for the area of a triangle:

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

Here, the area is 40, the height $AB$ is 10, and the base is $BC$ :

$40 = \frac{1}{2} \times BC \times 10$

We can simplify this equation to solve for $BC$ :

$40 = 5 \times BC$
$BC = \frac{40}{5}$
$BC = 8$

Hence, the length of side $BC$ is $\boxed{8}$ .

Final Answer

Key Points to Remember

Essential concepts to master this topic

Area Formula: Area equals half times base times height
Technique: Substitute known values: 40 = (1/2) × BC × 10
Check: Verify calculation: (1/2) × 8 × 10 = 40 ✓

Common Mistakes

Avoid these frequent errors

Confusing which sides are base and height
Don't assume any side is the base without checking the diagram = wrong measurements! The base and height must be perpendicular to each other in the triangle area formula. Always identify the right angle 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, the base and height are always the two sides that meet at the right angle. From the diagram, AB (length 10) and BC are perpendicular, so either can be base or height.

Why do we use the formula Area = (1/2) × base × height?

This formula works because a triangle is exactly half of a rectangle. If you draw a rectangle with the same base and height, the triangle takes up half the space!

What if I get a different answer when I switch base and height?

You should get the same answer either way! If AB = 10 is the base and BC is the height, or vice versa, the calculation $\frac{1}{2} \times 10 \times BC = 40$ gives BC = 8.

Can I use this method for any triangle?

This specific method works for right triangles where you know the area and one leg. For other triangles, you might need different formulas or additional information like angles.

How do I check if my answer makes sense?

Ask yourself: Does this create a reasonable triangle? With sides 8 and 10, and area 40, substitute back: $\frac{1}{2} \times 8 \times 10 = 40$ ✓

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