If `|{:(x_(1),y_(1),1),(x_(2),y_(2),1),(x_(3),y_(3),1):}|=|{:(a_(1),b_(1),1),(a_(2),b_(2),1),(a_(3),b_(3),1):}|` , then the two triangles with vertices `(x_(1),y_(1))`, `(x_(2),y_(2))`, `(x_(3),y_(3))` and `(a_(1),b_(1))`, `(a_(2),b_(2))` ,`(a_(3),b_(3))` must be congruent.
If `|{:(x_(1),y_(1),1),(x_(2),y_(2),1),(x_(3),y_(3),1):}|=|{:(a_(1),b_(1),1),(a_(2),b_(2),1),(a_(3),b_(3),1):}|` , then the two triangles with vertices `(x_(1),y_(1))`, `(x_(2),y_(2))`, `(x_(3),y_(3))` and `(a_(1),b_(1))`, `(a_(2),b_(2))` ,`(a_(3),b_(3))` must be congruent.
Text Solution
AI Generated Solution
The correct Answer is:
To determine whether the statement is true or false, we need to analyze the given condition regarding the areas of the triangles formed by the provided vertices.
### Step-by-Step Solution:
1. **Understanding the Given Condition**:
We are given two sets of points:
- Triangle 1 with vertices \((x_1, y_1)\), \((x_2, y_2)\), \((x_3, y_3)\)
- Triangle 2 with vertices \((a_1, b_1)\), \((a_2, b_2)\), \((a_3, b_3)\)
The condition states that the absolute value of the determinant formed by these points is equal:
\[
|{:(x_1, y_1, 1), (x_2, y_2, 1), (x_3, y_3, 1):}| = |{:(a_1, b_1, 1), (a_2, b_2, 1), (a_3, b_3, 1):}|
\]
2. **Area of the Triangle**:
The area of a triangle formed by the points \((x_1, y_1)\), \((x_2, y_2)\), \((x_3, y_3)\) can be calculated using the determinant:
\[
\text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|
\]
Similarly, the area for Triangle 2 can be calculated.
3. **Equal Areas**:
The condition given implies that the areas of both triangles are equal:
\[
\text{Area of Triangle 1} = \text{Area of Triangle 2}
\]
4. **Congruence of Triangles**:
For two triangles to be congruent, they must have:
- Equal corresponding sides
- Equal corresponding angles
However, having equal areas does not guarantee that the triangles are congruent. Triangles can have the same area but different shapes (e.g., different side lengths and angles).
5. **Conclusion**:
Since it is possible for two triangles to have equal areas without being congruent, the statement that the two triangles must be congruent is **false**.
### Final Answer:
The statement is **false**.
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