Evaluating the Alternative Area Expression

Oct 13, 2025 by windailys.com 110 views

Recall that in a $$45^{\circ}-45{\circ}-90^{\circ}$$ triangle, if the legs each measure $$x$$ units, then the hypotenuse measures $$x \sqrt{2}$$ units.

Choosing the Correct Area Expression in a $$45^{\circ}-45{\circ}-90^{\circ}$$ Triangle

A $$45^{\circ}-45{\circ}-90^{\circ}$$ triangle is a special right-angled triangle with two equal angles, each measuring $$45^{\circ}$$. This type of triangle has several unique properties, including the relationship between the lengths of its sides. Recall that in a $$45^{\circ}-45{\circ}-90^{\circ}$$ triangle, if the legs each measure $$x$$ units, then the hypotenuse measures $$x \sqrt{2}$$ units.

When working with a $$45^{\circ}-45\circ}-90^{\circ}$$ triangle, it's essential to understand how to calculate its area. The area of a triangle can be calculated using the formula Area = (base × height) / 2. In the case of a $$45^{\circ-45^{\circ}-90{\circ}$$ triangle, both the base and the height are equal to the length of the legs, which is $$x$$ units.

However, the area expression can be simplified using the properties of a $$45^{\circ}-45{\circ}-90^{\circ}$$ triangle. Since the legs are equal, we can use the formula: Area = (leg × leg) / 2. This simplifies to: Area = (x × x) / 2 = x^2 / 2.

Now, let's consider the given problem. We are asked to choose the correct area expression for a $$45^{\circ}-45{\circ}-90^{\circ}$$ triangle with legs measuring 12 units each. To find the area, we can use the formula: Area = (x × x) / 2 = x^2 / 2.

Calculating the Area of a $$45^{\circ}-45{\circ}-90^{\circ}$$ Triangle

To calculate the area of a $$45^{\circ}-45{\circ}-90^{\circ}$$ triangle, we need to substitute the value of x into the formula: Area = x^2 / 2. In this case, x = 12 units.

Area = (12 × 12) / 2 Area = 144 / 2 Area = 72

However, the problem statement provides an alternative area expression: (72 π - 144). To determine which expression is correct, we need to evaluate both expressions and compare the results.

Evaluating the Alternative Area Expression

The alternative area expression is: (72 π - 144). To evaluate this expression, we need to substitute the value of π (approximately 3.14) and perform the arithmetic operations.

First, we multiply 72 by π: 72 × 3.14 = 226.08

Next, we subtract 144 from the result: 226.08 - 144 = 82.08

Now, we have the result of the alternative area expression: 82.08. However, this result is not equal to the area of the triangle, which we calculated earlier as 72.

Choosing the Correct Area Expression

Based on our calculations, we can conclude that the correct area expression for a $$45^{\circ}-45{\circ}-90^{\circ}$$ triangle with legs measuring 12 units each is: x^2 / 2 = 72.

The alternative area expression: (72 π - 144) is incorrect, as it yields a result that is not equal to the area of the triangle.

Conclusion

In conclusion, when working with a $$45^{\circ}-45{\circ}-90^{\circ}$$ triangle, it's essential to understand the properties of this special right-angled triangle, including the relationship between the lengths of its sides. By using the correct area expression, we can calculate the area of the triangle accurately.

In this article, we have demonstrated how to choose the correct area expression for a $$45^{\circ}-45{\circ}-90^{\circ}$$ triangle. We have also evaluated an alternative area expression and shown that it is incorrect.

Final Answer

The final answer is: $\boxed{72}$
Q&A: Recall that in a $$45^{\circ}-45{\circ}-90^{\circ}$$ triangle, if the legs each measure $$x$$ units, then the hypotenuse measures $$x \sqrt{2}$$ units.

Frequently Asked Questions about $$45^{\circ}-45{\circ}-90^{\circ}$$ Triangles

In this article, we will answer some frequently asked questions about $$45^{\circ}-45{\circ}-90^{\circ}$$ triangles.

Q: What is the relationship between the lengths of the sides of a $$45^{\circ}-45{\circ}-90^{\circ}$$ triangle?

A: In a $$45^{\circ}-45{\circ}-90^{\circ}$$ triangle, the legs are equal, and the hypotenuse is $$\sqrt{2}$$ times the length of each leg.

Q: How do I calculate the area of a $$45^{\circ}-45{\circ}-90^{\circ}$$ triangle?

A: To calculate the area of a $$45^{\circ}-45{\circ}-90^{\circ}$$ triangle, you can use the formula: Area = (x × x) / 2 = x^2 / 2, where x is the length of each leg.

Q: What is the formula for the hypotenuse of a $$45^{\circ}-45{\circ}-90^{\circ}$$ triangle?

A: The formula for the hypotenuse of a $$45^{\circ}-45{\circ}-90^{\circ}$$ triangle is: hypotenuse = x √2, where x is the length of each leg.

Q: Can I use the Pythagorean theorem to calculate the area of a $$45^{\circ}-45{\circ}-90^{\circ}$$ triangle?

A: No, you cannot use the Pythagorean theorem to calculate the area of a $$45^{\circ}-45{\circ}-90^{\circ}$$ triangle. The Pythagorean theorem is used to calculate the length of the hypotenuse, not the area of the triangle.

Q: What is the relationship between the angles of a $$45^{\circ}-45{\circ}-90^{\circ}$$ triangle?

A: In a $$45^{\circ}-45{\circ}-90^{\circ}$$ triangle, two of the angles are equal, each measuring $$45^{\circ}$$. The third angle is $$90^{\circ}$$.

Q: Can I use a $$45^{\circ}-45{\circ}-90^{\circ}$$ triangle to solve real-world problems?

A: Yes, you can use a $$45^{\circ}-45{\circ}-90^{\circ}$$ triangle to solve real-world problems, such as calculating the area of a room or the length of a diagonal.

Conclusion

In conclusion, $$45^{\circ}-45{\circ}-90^{\circ}$$ triangles are special right-angled triangles with unique properties. By understanding the relationship between the lengths of its sides and the formula for the area, you can use this type of triangle to solve real-world problems.

Final Answer

The final answer is: $\boxed{72}$