Right Triangle Problem: Calculate Side Length Using Height 10cm and Area 40cm²

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

The base is the side that the height is perpendicular to. In this problem, height GE is perpendicular to side DF, so DF is our base.

Q: Why is the height 10 cm when GE looks diagonal?

Height in geometry means the perpendicular distance from a vertex to the opposite side. Even if GE looks slanted in the diagram, it represents the perpendicular height of 10 cm.

Q: Can I use a different formula for triangle area?

Yes! You could use $ \text{Area} = \frac{1}{2}ab\sin C $ if you know two sides and the included angle, but the base-height formula is simpler when height is given directly.

Q: How can I check if my answer makes sense?

Ask yourself: Does this length seem reasonable? If the area is 40 cm² and height is 10 cm, a base of 8 cm gives us a triangle that's not too skinny or too wide - that makes sense!

Triangle Area Formula with Height-Base Relationships

DEF is a right triangle.

Height GE is 10 cm.
The area of DEF is 40 cm².

Calculate the length of side DF.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the line DF

00:03 Identify the size of the sides and lines according to the given data

00:07 Apply the formula for calculating the area of a triangle

00:10 (height x base) divided by 2

00:15 Substitute in the relevant values and proceed to solve for DF

00:24 Multiply by the denominators in order to eliminate fractions

00:31 Isolate DF

00:46 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

DEF is a right triangle.

Height GE is 10 cm.
The area of DEF is 40 cm².

Calculate the length of side DF.

Step-by-step solution

To solve this problem, we will find the length of side DF using the formula for the area of a triangle:

The area of a triangle is given by:
$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$

For triangle DEF, the area is given as 40 cm², and the height GE is 10 cm. We can consider side DF as the base. Therefore, substitute the given values:
$\frac{1}{2} \times \text{DF} \times 10 = 40$

Simplify this expression:
$\text{DF} \times 5 = 40$

Divide both sides by 5 to solve for DF:
$\text{DF} = \frac{40}{5} = 8$

Thus, the length of side DF is $\text{8 cm}$ .

By comparing with the given choices, the correct answer is indeed choice 1, which is 8 cm.

Therefore, the solution to the problem is $\text{8 cm}$ .

Final Answer

8 cm

Key Points to Remember

Essential concepts to master this topic

Area Formula: Area = $\frac{1}{2} \times \text{base} \times \text{height}$
Technique: Substitute known values: $\frac{1}{2} \times DF \times 10 = 40$
Check: Verify: $\frac{1}{2} \times 8 \times 10 = 40$ cm² ✓

Common Mistakes

Avoid these frequent errors

Using wrong height-base relationship
Don't assume any side can be the base without checking the height = wrong calculation! The height must be perpendicular to the base you choose. Always identify which line segment is the height and which side it's perpendicular to.

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 side is the base?

The base is the side that the height is perpendicular to. In this problem, height GE is perpendicular to side DF, so DF is our base.

Why is the height 10 cm when GE looks diagonal?

Height in geometry means the perpendicular distance from a vertex to the opposite side. Even if GE looks slanted in the diagram, it represents the perpendicular height of 10 cm.

Can I use a different formula for triangle area?

Yes! You could use $\text{Area} = \frac{1}{2}ab\sin C$ if you know two sides and the included angle, but the base-height formula is simpler when height is given directly.

What if I get a decimal answer?

That's perfectly normal! Many geometry problems have decimal solutions. Just make sure to include the correct units (cm, m, etc.) in your final answer.

How can I check if my answer makes sense?

Ask yourself: Does this length seem reasonable? If the area is 40 cm² and height is 10 cm, a base of 8 cm gives us a triangle that's not too skinny or too wide - that makes sense!

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