What is the area of the triangle formed by points (a, c + a), (a, c) and (–a, c − a) ?

Q: What is the area of the triangle formed by points (a, c + a), (a, c) and (–a, c − a) ?

To determine the area of the triangle formed by the points $(a, c + a)$, $(a, c)$, and $(-a, c - a)$, we can use the formula for the area of a triangle given its vertices $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$: \[ \text{Area} = \frac{1}{2} \left| x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) \right| \] Substituting the given points into the formula: - $(x_1, y_1) = (a, c + a)$ - $(x_2, y_2) = (a, c)$ - $(x_3, y_3) = (-a, c - a)$ \[ \text{Area} = \frac{1}{2} \left| a(c - (c - a)) + a((c - a) - (c + a)) + (-a)((c + a) - c) \right| \] Simplifying each term: 1. $a(c - (c - a)) = a(a) = a^2$ 2. $a((c - a) - (c + a)) = a(-2a) = -2a^2$ 3. $(-a)((c + a) - c) = (-a)(a) = -a^2$ Putting it all together: \[ \text{Area} = \frac{1}{2} \left| a^2 - 2a^2 - a^2 \right| = \frac{1}{2} \left| -2a^2 \right| = \frac{1}{2} \times 2a^2 = a^2 \] Thus, the area of the triangle is $a^2$, which corresponds to Option 2. **Explanation of Incorrect Options:** 1. **Option 1: 1** - This option is incorrect because the area of the triangle depends on the variable $a$, and it cannot be a constant value like 1 unless specific conditions are met, which are not given here. 3. **Option 3: $\sqrt{a^2+c^2}$** - This option is incorrect because $\sqrt{a^2+c^2}$ represents a distance, not an area. The area of a triangle is not calculated using the Euclidean distance formula. 4. **Option 4: $\sqrt{c^2-a^2}$** - This option is incorrect because, similar to Option 3, it represents a distance or a length, not an area. The area of a triangle cannot be expressed as the square root of a difference of squares unless specific geometric conditions apply, which are not present here. Therefore, the correct answer is indeed Option 2: $a^2$.

Question

Accepted Answer

To determine the area of the triangle formed by the points $(a, c + a)$, $(a, c)$, and $(-a, c - a)$, we can use the formula for the area of a triangle given its vertices $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$:

$$
\text{Area} = \frac{1}{2} \left| x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) \right|
$$

Substituting the given points into the formula:

- $(x_1, y_1) = (a, c + a)$
- $(x_2, y_2) = (a, c)$
- $(x_3, y_3) = (-a, c - a)$

$$
\text{Area} = \frac{1}{2} \left| a(c - (c - a)) + a((c - a) - (c + a)) + (-a)((c + a) - c) \right|
$$

Simplifying each term:

1. $a(c - (c - a)) = a(a) = a^2$
2. $a((c - a) - (c + a)) = a(-2a) = -2a^2$
3. $(-a)((c + a) - c) = (-a)(a) = -a^2$

Putting it all together:

$$
\text{Area} = \frac{1}{2} \left| a^2 - 2a^2 - a^2 \right| = \frac{1}{2} \left| -2a^2 \right| = \frac{1}{2} \times 2a^2 = a^2
$$

Thus, the area of the triangle is $a^2$, which corresponds to Option 2.

**Explanation of Incorrect Options:**

1. **Option 1: 1** - This option is incorrect because the area of the triangle depends on the variable $a$, and it cannot be a constant value like 1 unless specific conditions are met, which are not given here.

3. **Option 3: $\sqrt{a^2+c^2}$** - This option is incorrect because $\sqrt{a^2+c^2}$ represents a distance, not an area. The area of a triangle is not calculated using the Euclidean distance formula.

4. **Option 4: $\sqrt{c^2-a^2}$** - This option is incorrect because, similar to Option 3, it represents a distance or a length, not an area. The area of a triangle cannot be expressed as the square root of a difference of squares unless specific geometric conditions apply, which are not present here.

Therefore, the correct answer is indeed Option 2: $a^2$.