Right Triangle Challenge: Find Side Length Given Area 10.5 and Height 3

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 3) and BC are the legs, so they're your base and height. Never use the hypotenuse AC.

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

A triangle is exactly half of a rectangle! If you draw a rectangle with the same base and height, the triangle's area is half of that rectangle's area.

Q: Can I use this formula for any triangle?

No! This formula $ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} $ works for all triangles, but the height must be perpendicular to the base. In right triangles, the legs are already perpendicular.

Q: How do I know if my final answer makes sense?

Ask yourself: Does this create a reasonable triangle? With legs of 3 and 7, you get area = ½ × 3 × 7 = 10.5, which matches the given information!

Triangle Area Formula with Given Measurements

A right triangle is shown below.

Its area is 10.5.

Calculate the length of side BC.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:07 Let's find the value of BC.

00:10 We'll use the formula for the area of a triangle.

00:13 Area equals height A B times base B C, then divided by 2.

00:20 Now, substitute the values you know and calculate BC.

00:25 Multiply the equation by 2 to get rid of the fraction.

00:31 Next, isolate BC on one side.

00:39 And there you have it! That's how you find BC.

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

A right triangle is shown below.

Its area is 10.5.

Calculate the length of side BC.

Step-by-step solution

To solve for the length of side $BC$ in the right triangle, we will use the area formula for triangles:

Step 1: Identify the given elements: $\text{Area} = 10.5$ and one leg $AB = 3$ .
Step 2: Use the formula for the area of a right triangle:
$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$
Step 3: Substitute the known values into the formula:
$10.5 = \frac{1}{2} \times 3 \times BC$
Step 4: Solve for $BC$ :
Multiply both sides by 2 to clear the fraction:
$21 = 3 \times BC$
Step 5: Divide both sides by 3 to isolate $BC$ :
$BC = \frac{21}{3} = 7$

Therefore, the length of side $BC$ is $\boxed{7}$ .

Final Answer

Key Points to Remember

Essential concepts to master this topic

Area Formula: For right triangles, Area = ½ × base × height
Technique: Substitute known values: 10.5 = ½ × 3 × BC
Check: Verify by calculating: ½ × 3 × 7 = 10.5 ✓

Common Mistakes

Avoid these frequent errors

Using wrong sides as base and height
Don't use the hypotenuse as base or height = wrong calculation! The area formula only works with the two perpendicular sides (legs) of the right triangle. Always identify the two legs that form the right angle as your base and height.

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 3) and BC are the legs, so they're your base and height. Never use the hypotenuse AC.

Why do we multiply by ½ in the area formula?

A triangle is exactly half of a rectangle! If you draw a rectangle with the same base and height, the triangle's area is half of that rectangle's area.

What if I get a decimal answer instead of 7?

Check your arithmetic! With area 10.5 and one leg of 3, the calculation should give you exactly 7. Double-check each step: 10.5 × 2 = 21, then 21 ÷ 3 = 7.

Can I use this formula for any triangle?

No! This formula $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$ works for all triangles, but the height must be perpendicular to the base. In right triangles, the legs are already perpendicular.

How do I know if my final answer makes sense?

Ask yourself: Does this create a reasonable triangle? With legs of 3 and 7, you get area = ½ × 3 × 7 = 10.5, which matches the given information!

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