Calculate the Area Function: Shaded Triangle in a Square with Side Length a

Q: How do I know the base and height of this triangle?

When a diagonal creates a triangle in a square, both the base and height equal the side length of the square. The diagonal goes from one corner to the opposite corner, making two sides of the square into the triangle's base and height.

Q: Why isn't the answer just a²?

The area $ a^2 $ is for the entire square! The shaded triangle is only half of the square, so we need $ \frac{1}{2} \times a^2 = \frac{a^2}{2} $.

Q: What if the triangle looks different in the diagram?

As long as the triangle is formed by a diagonal of the square, it will always have area $ \frac{a^2}{2} $. The diagonal creates a right triangle with legs of length a.

Q: How can I double-check my answer?

Remember that two identical triangles should make up the whole square. So $ 2 \times \frac{a^2}{2} = a^2 $, which is indeed the square's area!

Q: Does it matter which corner the diagonal starts from?

No! Any diagonal in a square creates two congruent right triangles, each with the same area $ \frac{a^2}{2} $.

Triangle Area with Square Diagonal Formation

A square has a side length of a.

Choose the function that expresses the area of the shaded triangle.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Express the shaded area

00:03 In a square all sides are equal

00:07 We'll use the formula for calculating triangle area

00:10 (height times side) divided by 2

00:13 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

A square has a side length of a.

Choose the function that expresses the area of the shaded triangle.

Step-by-step solution

To solve the problem, we will calculate the area of the triangle within a square of side length $a$ .

Since the triangle is formed by a diagonal of the square, its base and height are both equal to the side length $a$ of the square. Thus, base = $a$ and height = $a$ .

Using the formula for the area of a triangle, we have:

$\text{Area of the triangle} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times a \times a = \frac{1}{2} \times a^2$

This simplifies to $\frac{a^2}{2}$ .

Therefore, the function that expresses the area of the shaded triangle is $\frac{a^2}{2}$ , matching the choice with the answer: $y=\frac{a^2}{2}$ .

The solution to the problem is $y=\frac{a^2}{2}$ .

Final Answer

$y=\frac{a^2}{2}$

Key Points to Remember

Essential concepts to master this topic

Formula: Triangle area equals one-half times base times height
Technique: For diagonal triangles, both base and height equal side length a
Check: Diagonal splits square into two equal triangles, each with area $\frac{a^2}{2}$ ✓

Common Mistakes

Avoid these frequent errors

Using the full square area instead of triangle area
Don't calculate the entire square area $a^2$ as your answer! The triangle is only half the square, so this doubles the correct result. Always apply the triangle formula $\frac{1}{2} \times base \times height$ to get $\frac{a^2}{2}$ .

Practice Quiz

Test your knowledge with interactive questions

Complete:

The missing value of the function point:

$$ f(x)=x^2 $$

$$ f(?)=16 $$

FAQ

Everything you need to know about this question

How do I know the base and height of this triangle?

When a diagonal creates a triangle in a square, both the base and height equal the side length of the square. The diagonal goes from one corner to the opposite corner, making two sides of the square into the triangle's base and height.

Why isn't the answer just a²?

The area $a^2$ is for the entire square! The shaded triangle is only half of the square, so we need $\frac{1}{2} \times a^2 = \frac{a^2}{2}$ .

What if the triangle looks different in the diagram?

As long as the triangle is formed by a diagonal of the square, it will always have area $\frac{a^2}{2}$ . The diagonal creates a right triangle with legs of length a.

How can I double-check my answer?

Remember that two identical triangles should make up the whole square. So $2 \times \frac{a^2}{2} = a^2$ , which is indeed the square's area!

Does it matter which corner the diagonal starts from?

No! Any diagonal in a square creates two congruent right triangles, each with the same area $\frac{a^2}{2}$ .

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