Calculate Triangle Area: Using 1/4 Side Ratio and Height of 12

Q: What does 'BC is 1/4 side AE' actually mean?

This means BC's length equals one-fourth of AE's length. If AE = 12, then BC = $ \frac{1}{4} \times 12 = 3 $. It's not saying BC equals the fraction $ \frac{1}{4} $!

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

From the diagram, you can see that AE is perpendicular to BC (shown by the vertical line). This makes BC the base and AE the height. The height is always perpendicular to the base!

Q: Why isn't the answer just 12 since that's the height shown?

Area isn't just the height! You need both base and height to calculate area. The formula $ A = \frac{1}{2} \times \text{base} \times \text{height} $ requires multiplying base (3) by height (12), then dividing by 2.

Q: What if I calculated BC wrong from the ratio?

Double-check your ratio calculation: $ \frac{1}{4} \times 12 = \frac{12}{4} = 3 $. If you get BC = 3, then area = $ \frac{1}{2} \times 3 \times 12 = 18 $. Any other BC value means you need to recalculate the ratio step.

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:06 Let's calculate the area of this triangle.

00:12 We know that B C is one quarter of A E, based on the data given.

00:18 Next, we'll use the value of A E to find B C. Let's substitute it in.

00:25 Now, we will use the triangle area formula.

00:29 That's base, which is B C, times height, which is A E, divided by two.

00:36 Let's substitute the values to calculate the area.

00:44 When we divide twelve by two, we get six.

00:49 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 $\frac{1}{4}$ side AE.

Calculate the area of the triangle:

2

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Calculate the length of side BC using the given ratio to AE.
Step 2: Use the formula for the area of a triangle with the calculated BC and given AE.

Let's work through these steps:

Step 1: Calculate BC
Given that BC is $\frac{1}{4}$ the length of AE, and AE is 12, we find the length of BC as follows:

$BC = \frac{1}{4} \times 12 = 3$

Step 2: Use the formula for the area of a triangle
The formula is given by $A = \frac{1}{2} \times \text{base} \times \text{height}$ .

In this context, BC serves as the base, and AE serves as the height. Thus, we compute the area:

$A = \frac{1}{2} \times BC \times AE = \frac{1}{2} \times 3 \times 12 = \frac{1}{2} \times 36 = 18$

Therefore, the area of triangle $\triangle$ ABC is $\mathbf{18}$ .

Thus, the correct answer is choice 4: $\mathbf{18}$ .

3

Final Answer

18

Key Points to Remember

Essential concepts to master this topic

Ratio Rule: Convert side ratio to actual length by multiplication
Area Formula: $A = \frac{1}{2} \times \text{base} \times \text{height}$
Check: Verify base × height calculation: 3 × 12 = 36 ÷ 2 = 18 ✓

Common Mistakes

Avoid these frequent errors

Using the ratio fraction directly as the base length
Don't use $\frac{1}{4}$ as the base length = area of $\frac{1}{2} \times \frac{1}{4} \times 12 = 1.5$ ! The ratio tells you how to find the actual length. Always multiply the ratio by the reference length: $\frac{1}{4} \times 12 = 3$ to get the true base.

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

What does 'BC is 1/4 side AE' actually mean?

+

This means BC's length equals one-fourth of AE's length. If AE = 12, then BC = $\frac{1}{4} \times 12 = 3$ . It's not saying BC equals the fraction $\frac{1}{4}$ !

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

+

From the diagram, you can see that AE is perpendicular to BC (shown by the vertical line). This makes BC the base and AE the height. The height is always perpendicular to the base!

Why isn't the answer just 12 since that's the height shown?

+

Area isn't just the height! You need both base and height to calculate area. The formula $A = \frac{1}{2} \times \text{base} \times \text{height}$ requires multiplying base (3) by height (12), then dividing by 2.

Can I use a different pair of sides as base and height?

+

You could, but you'd need to know the perpendicular distance between them. In this problem, AE is clearly shown as the height perpendicular to base BC, making the calculation straightforward.

What if I calculated BC wrong from the ratio?

+

Double-check your ratio calculation: $\frac{1}{4} \times 12 = \frac{12}{4} = 3$ . If you get BC = 3, then area = $\frac{1}{2} \times 3 \times 12 = 18$ . Any other BC value means you need to recalculate the ratio step.

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Calculate Triangle Area: Using 1/4 Side Ratio and Height of 12

Triangle Area with Side Ratios

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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

What does 'BC is 1/4 side AE' actually mean?

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

Why isn't the answer just 12 since that's the height shown?

Can I use a different pair of sides as base and height?

What if I calculated BC wrong from the ratio?

🌟 Unlock Your Math Potential

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