Right Triangle Area Problem: Find BC When Area = 38 cm² and AC = 8 cm

Q: How do I know which sides are the base and height?

In a right triangle, the base and height are the two perpendicular sides that form the right angle. The hypotenuse (longest side) is never used in the area formula.

Q: What if I don't know which angle is the right angle?

Look for the right angle symbol (small square) in the diagram! The two sides that meet at this square corner are your base and height.

Q: Why do we multiply by ½ in the area formula?

A right triangle is exactly half of a rectangle. Since a rectangle's area is base × height, the triangle's area is $ \frac{1}{2} \times \text{base} \times \text{height} $.

Q: What if my answer doesn't match any of the given choices?

Double-check your arithmetic! Make sure you multiplied both sides by 2 correctly and divided properly. Also verify you used the right area formula.

Q: How can I be sure BC = 9.5 cm is correct?

Substitute back: $ \frac{1}{2} \times 8 \times 9.5 = 4 \times 9.5 = 38 $ cm². Since this matches the given area, your answer is correct!

Right Triangle Area with Given Side

The triangle ABC is a right triangle.

The area of the triangle is 38 cm².

AC = 8

Calculate side BC.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:10 Let's find the height, B C.

00:13 First, check the sizes of the sides and lines given.

00:18 Next, use the formula for the area of a triangle.

00:26 That's, height times base, divided by two.

00:30 Now, plug in the values to find B C.

00:46 Multiply to clear fractions.

01:02 Isolate B C.

01:15 And that's your solution!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

The triangle ABC is a right triangle.

The area of the triangle is 38 cm².

AC = 8

Calculate side BC.

Step-by-step solution

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

Step 1: Identify the area formula related to a right triangle.
Step 2: Set up an equation with the given area and known side.
Step 3: Solve for the unknown side $BC$ .

Step 1: We know the area $A$ of a right triangle is given by the formula:
$A = \frac{1}{2} \times \text{base} \times \text{height}$ .

Step 2: Using the known values, $A = 38 \, \text{cm}^2$ , the side $\text{AC} = 8 \, \text{cm}$ and assuming it acts as the base, we set up the equation:
$38 = \frac{1}{2} \times 8 \times \text{BC}$ .

Step 3: Simplify to solve for $\text{BC}$ :
Multiply both sides by 2 to eliminate the fraction:
$76 = 8 \times \text{BC}$ .
Now, divide both sides by 8 to find $\text{BC}$ :
$\text{BC} = \frac{76}{8}$ .
$\text{BC} = 9.5 \, \text{cm}$ .

Therefore, the length of side $BC$ is 9.5 cm.

Final Answer

9.5 cm

Key Points to Remember

Essential concepts to master this topic

Formula: Area of right triangle equals one-half base times height
Technique: Substitute known values: 38 = ½ × 8 × BC
Check: Verify by calculating: ½ × 8 × 9.5 = 38 cm² ✓

Common Mistakes

Avoid these frequent errors

Using the wrong area formula
Don't use Area = ½ × hypotenuse × other side = incorrect result! The hypotenuse is never used in the area formula for right triangles. Always use Area = ½ × base × height where base and height are the perpendicular sides.

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

How do I know which sides are the base and height?

In a right triangle, the base and height are the two perpendicular sides that form the right angle. The hypotenuse (longest side) is never used in the area formula.

What if I don't know which angle is the right angle?

Look for the right angle symbol (small square) in the diagram! The two sides that meet at this square corner are your base and height.

Can I use any side as the base?

Yes! You can choose either perpendicular side as the base. The other perpendicular side becomes the height. Just never use the hypotenuse in the area formula.

Why do we multiply by ½ in the area formula?

A right triangle is exactly half of a rectangle. Since a rectangle's area is base × height, the triangle's area is $\frac{1}{2} \times \text{base} \times \text{height}$ .

What if my answer doesn't match any of the given choices?

Double-check your arithmetic! Make sure you multiplied both sides by 2 correctly and divided properly. Also verify you used the right area formula.

How can I be sure BC = 9.5 cm is correct?

Substitute back: $\frac{1}{2} \times 8 \times 9.5 = 4 \times 9.5 = 38$ cm². Since this matches the given area, your answer is correct!

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