Right Triangle Side Length: Finding BC When Area = 10 and Height = 4

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

In a right triangle, the base and height are the two sides that form the right angle. They're perpendicular to each other. The hypotenuse (longest side) is never used as base or height in the area formula.

Q: Can I use any two sides for base and height?

No! You can only use the two sides that are perpendicular (form a 90° angle). In this problem, AB and BC are perpendicular, so AB = 4 is the height and BC is what we're solving for.

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

You should get the same area either way! If AB = 4 is base and BC = 5 is height: $ \frac{1}{2} \times 4 \times 5 = 10 $. The area formula works both ways.

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

A 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. That's why we multiply by $ \frac{1}{2} $.

Q: Can this method work for any triangle?

This $ \frac{1}{2} \times \text{base} \times \text{height} $ formula works for all triangles, not just right triangles! You just need to identify a true height (perpendicular distance from base to opposite vertex).

Triangle Area Formula with Given Height

Look at the right triangle below.

Area = 10

How long is side BC?

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

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 calculate the triangle's area

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

00:11 Substitute in the relevant values and calculate to find BC

00:20 Divide 4 by 2 to obtain 2

00:23 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

Look at the right triangle below.

Area = 10

How long is side BC?

Step-by-step solution

To find the length of side $BC$ , follow these steps:

Step 1: Identify the given information

The area of the triangle is given as $10$ .
The height $AB$ is $4$ .

Step 2: Apply the area formula for a right triangle

The formula for the area $A$ of a triangle is:

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

Step 3: Set up the equation

Substituting the known values into the formula gives:

$10 = \frac{1}{2} \times BC \times 4$

Step 4: Solve for $BC$

Begin by simplifying the equation:

$10 = 2 \times BC$

Dividing both sides by 2 to solve for $BC$ , we obtain:

$BC = \frac{10}{2} = 5$

Therefore, the length of side $BC$ is $5$ .

Final Answer

Key Points to Remember

Essential concepts to master this topic

Area Formula: Area = $\frac{1}{2} \times \text{base} \times \text{height}$ for triangles
Technique: Substitute known values: 10 = $\frac{1}{2} \times BC \times 4$
Check: Verify: $\frac{1}{2} \times 5 \times 4 = 10$ ✓

Common Mistakes

Avoid these frequent errors

Confusing which sides are base and height
Don't assume the longest side is the base = wrong calculation! In right triangles, any two perpendicular sides can be base and height. Always identify the right angle first, then 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 the two sides that form the right angle. They're perpendicular to each other. The hypotenuse (longest side) is never used as base or height in the area formula.

Can I use any two sides for base and height?

No! You can only use the two sides that are perpendicular (form a 90° angle). In this problem, AB and BC are perpendicular, so AB = 4 is the height and BC is what we're solving for.

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

You should get the same area either way! If AB = 4 is base and BC = 5 is height: $\frac{1}{2} \times 4 \times 5 = 10$ . The area formula works both ways.

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

A 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. That's why we multiply by $\frac{1}{2}$ .

Can this method work for any triangle?

This $\frac{1}{2} \times \text{base} \times \text{height}$ formula works for all triangles, not just right triangles! You just need to identify a true height (perpendicular distance from base to opposite vertex).

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Triangle questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime