Calculate BC Length in Right Triangle: Area = 7, Side = 2

Q: Which sides should I use as base and height?

Use the two perpendicular sides that form the right angle! In this triangle, AB (length 2) and BC are perpendicular, so they're your base and height.

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

The area formula $ A = \frac{1}{2} \times \text{base} \times \text{height} $ requires perpendicular sides. The hypotenuse is never perpendicular to another side in a right triangle.

Q: How do I know which vertex has the right angle?

Look for the square symbol or check which angle measures 90°. In this problem, the right angle is at vertex B, so sides AB and BC are perpendicular.

Q: Can I solve this using other triangle formulas?

While other formulas exist, the basic area formula $ A = \frac{1}{2} \times \text{base} \times \text{height} $ is the most direct approach when you know the area and one perpendicular side.

Q: What if I get the same answer using different sides?

That's impossible! Only the correct perpendicular sides will give you the right answer. If you get the same result with different sides, double-check your triangle orientation.

Triangle Area Formula with Side Length Finding

ABC is a right triangle with an area of of 7.

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 area of a triangle

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

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

00:18 Multiply by 2 to avoid fractions

00:26 Isolate BC

00:34 Reduce the 2

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 of 7.

Calculate the length of side BC.

Step-by-step solution

To solve this problem, let's apply the formula for the area of a triangle:

The area $A$ of a right triangle can be expressed as

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

Let's denote side BC (the base) as $b$ and the given height AB as 2. Substituting in the known values, we have:

7 = \frac{1}{2} \times b \times 2

Simplifying the right side gives us:

7 = b

Therefore, the length of side BC is 7 units.

Final Answer

Key Points to Remember

Essential concepts to master this topic

Rule: Right triangle area equals half times base times height
Technique: Substitute known values: $7 = \frac{1}{2} \times BC \times 2$
Check: Verify with area formula: $\frac{1}{2} \times 7 \times 2 = 7$ ✓

Common Mistakes

Avoid these frequent errors

Using the wrong sides as base and height
Don't assume any two sides can be base and height = wrong calculation! Only perpendicular sides (forming the right angle) work as base and height. Always identify which sides are perpendicular in your right triangle.

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

Which sides should I use as base and height?

Use the two perpendicular sides that form the right angle! In this triangle, AB (length 2) and BC are perpendicular, so they're your base and height.

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

The area formula $A = \frac{1}{2} \times \text{base} \times \text{height}$ requires perpendicular sides. The hypotenuse is never perpendicular to another side in a right triangle.

How do I know which vertex has the right angle?

Look for the square symbol or check which angle measures 90°. In this problem, the right angle is at vertex B, so sides AB and BC are perpendicular.

Can I solve this using other triangle formulas?

While other formulas exist, the basic area formula $A = \frac{1}{2} \times \text{base} \times \text{height}$ is the most direct approach when you know the area and one perpendicular side.

What if I get the same answer using different sides?

That's impossible! Only the correct perpendicular sides will give you the right answer. If you get the same result with different sides, double-check your triangle orientation.

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