Right Triangle Side Length: Finding BC When Area = 21

Q: Why can't I use the hypotenuse in the area formula?

The area formula $ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} $ requires two perpendicular sides. The hypotenuse is slanted, so it doesn't form a right angle with either leg.

Q: How do I know which sides are the legs?

The legs are the two sides that meet at the right angle (90°). In this diagram, sides AB and BC are legs because they form the right angle at point B.

Q: What if I get a decimal answer?

Check your arithmetic carefully! In this problem, $ \frac{42}{7} = 6 $ exactly. If you get a decimal, you might have made an error in your calculation.

Q: Why do I multiply both sides by 2?

To isolate the product $ 7 \times BC $. When you have $ 21 = \frac{1}{2} \times 7 \times BC $, multiplying by 2 eliminates the fraction: $ 42 = 7 \times BC $.

Area Formula with Given Side Length

ABC is a right triangle with an area of 21.

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 for calculating the triangle's area

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

00:10 Substitute in the relevant values and calculate to determine BC

00:17 Multiply by 2 to avoid fractions

00:24 Isolate BC

00:31 Divide 21 by 7 to obtain 3

00:36 That's 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 21.

Calculate the length of side BC.

Step-by-step solution

To solve the problem, we start by identifying that the area of a right triangle is given by the formula:

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

Given: $\text{Area} = 21$ and one leg of the triangle, say the height $AB = 7$ .

We denote the other leg, which we need to find, as $BC$ . Thus:

$21 = \frac{1}{2} \times 7 \times BC$

Solving for $BC$ , first multiply both sides by 2 to isolate the product of $7$ and $BC$ :

$42 = 7 \times BC$

Now, divide both sides by 7 to solve for $BC$ :

$BC = \frac{42}{7} = 6$

Therefore, the length of side $BC$ is $\mathbf{6}$ .

Final Answer

Key Points to Remember

Essential concepts to master this topic

Area Formula: Right triangle area equals one-half times base times height
Technique: Substitute known values: $21 = \frac{1}{2} \times 7 \times BC$
Check: Verify: $\frac{1}{2} \times 7 \times 6 = 21$ ✓

Common Mistakes

Avoid these frequent errors

Using the wrong legs in the area formula
Don't use the hypotenuse in the area formula = wrong calculation! The area formula only uses the two perpendicular sides (legs), not the slanted side. Always identify which sides form the right angle before applying the formula.

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

Why can't I use the hypotenuse in the area formula?

The area formula $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$ requires two perpendicular sides. The hypotenuse is slanted, so it doesn't form a right angle with either leg.

How do I know which sides are the legs?

The legs are the two sides that meet at the right angle (90°). In this diagram, sides AB and BC are legs because they form the right angle at point B.

What if I get a decimal answer?

Check your arithmetic carefully! In this problem, $\frac{42}{7} = 6$ exactly. If you get a decimal, you might have made an error in your calculation.

Can I solve this using the Pythagorean theorem instead?

Not directly! The Pythagorean theorem finds missing sides when you know two sides, but here we only know one side and the area. Always use the area formula first to find the missing leg.

Why do I multiply both sides by 2?

To isolate the product $7 \times BC$ . When you have $21 = \frac{1}{2} \times 7 \times BC$ , multiplying by 2 eliminates the fraction: $42 = 7 \times BC$ .

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