Triangle Area Calculation: Using 1/3 Ratio and 6-Unit Height

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

The base can be any side of the triangle, but the height must be perpendicular to that base. In this problem, AE is drawn as a vertical line from A to the base BC, making it the height.

Q: What does the ratio 1/3 actually tell me?

The ratio $ \frac{1}{3} $ means BC is one-third the length of AE. So if AE = 6, then $ BC = \frac{1}{3} \times 6 = 2 $.

Q: Why do I multiply by 1/2 in the area formula?

The triangle area formula $ \frac{1}{2} \times \text{base} \times \text{height} $ comes from the fact that a triangle is half of a rectangle with the same base and height.

Q: What if the ratio was different, like 2/5?

The same method applies! If $ BC = \frac{2}{5} \times AE $ and AE = 6, then $ BC = \frac{2}{5} \times 6 = \frac{12}{5} = 2.4 $.

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:06 Let's find the area of the triangle.

00:11 BC is one-third of AE, as given in the problem.

00:16 Now, substitute AE's value to find BC.

00:24 We'll use the formula for a triangle's area.

00:28 That's base BC times height AE, divided by 2.

00:33 Substitute the values you have and calculate to find the area, X.

00:38 Don't forget, you can reduce by 2.

00:42 And there you have it. That's the solution!

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

Since the side BC is $\frac{1}{3}$ side AE.

Calculate the area of the triangle:

2

Step-by-step solution

Let's calculate the area of triangle ABC by following these steps:

Step 1: Calculate AE using the ratio. Given that side BC is $\frac{1}{3}$ of AE, if we call AE = 6 (as shown or suggested by the illustration, considering AE is the vertical height), then $BC = \frac{1}{3} \times 6 = 2$ .
Step 2: The height of triangle ABC is given by the length AE, which is 6.
Step 3: Calculate the area using the formula for the area of a triangle: $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times BC \times AE = \frac{1}{2} \times 2 \times 6 = 6$ .

Therefore, the area of triangle ABC is $\text{6}$ .

3

Final Answer

6

Key Points to Remember

Essential concepts to master this topic

Formula: Triangle area equals one-half base times height
Technique: Use ratio $BC = \frac{1}{3} \times AE = \frac{1}{3} \times 6 = 2$
Check: Area = $\frac{1}{2} \times 2 \times 6 = 6$ ✓

Common Mistakes

Avoid these frequent errors

Using wrong measurements for base and height
Don't assume BC is the height just because it's given = wrong calculation! The ratio tells us BC is the base (length 2) while AE is the perpendicular height (length 6). Always identify which measurement is the base and which is the perpendicular height.

Practice Quiz

Test your knowledge with interactive questions

What is the ratio between the orange and gray parts in the drawing?

FAQ

Everything you need to know about this question

How do I know which side is the base and which is the height?

+

The base can be any side of the triangle, but the height must be perpendicular to that base. In this problem, AE is drawn as a vertical line from A to the base BC, making it the height.

What does the ratio 1/3 actually tell me?

+

The ratio $\frac{1}{3}$ means BC is one-third the length of AE. So if AE = 6, then $BC = \frac{1}{3} \times 6 = 2$ .

Why do I multiply by 1/2 in the area formula?

+

The triangle area formula $\frac{1}{2} \times \text{base} \times \text{height}$ comes from the fact that a triangle is half of a rectangle with the same base and height.

Can I use a different side as the base?

+

Yes! You can use any side as the base, but then you need to find the perpendicular height to that base. The area will be the same regardless of which base you choose.

What if the ratio was different, like 2/5?

+

The same method applies! If $BC = \frac{2}{5} \times AE$ and AE = 6, then $BC = \frac{2}{5} \times 6 = \frac{12}{5} = 2.4$ .

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Triangle Area Calculation: Using 1/3 Ratio and 6-Unit Height

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Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

How do I know which side is the base and which is the height?

What does the ratio 1/3 actually tell me?

Why do I multiply by 1/2 in the area formula?

Can I use a different side as the base?

What if the ratio was different, like 2/5?

🌟 Unlock Your Math Potential

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