Question
Point A lies at (2,4) and point B lies at (3,6) . Which point is needed so that △ABC is isosceles?(1 point)ResponsesC(5,4)upper C left parenthesis 5 comma 4 right parenthesisC(5,5)upper C left parenthesis 5 comma 5 right parenthesisC(3,4)upper C left parenthesis 3 comma 4 right parenthesisC(0,4)
Answers
GPT-4o mini
To determine which point \( C \) will make triangle \( ABC \) isosceles, we can calculate the distances from point \( C \) to points \( A \) and \( B \) and check for equality.
Let's denote the coordinates of the points:
- \( A(2, 4) \)
- \( B(3, 6) \)
- \( C(x, y) \) (where \( (x, y) \) will be one of the options provided)
We can compute the squared distances to avoid dealing with square roots:
1. **Distance from \( A \) to \( B \)**:
\[
AB^2 = (3-2)^2 + (6-4)^2 = 1^2 + 2^2 = 1 + 4 = 5
\]
Now, we will check each option for point \( C \) to see if it results in two equal distances:
1. **For \( C(5, 4) \)**:
\[
AC^2 = (5-2)^2 + (4-4)^2 = 3^2 + 0^2 = 9 + 0 = 9
\]
\[
BC^2 = (5-3)^2 + (4-6)^2 = 2^2 + (-2)^2 = 4 + 4 = 8
\]
(Not isosceles)
2. **For \( C(5, 5) \)**:
\[
AC^2 = (5-2)^2 + (5-4)^2 = 3^2 + 1^2 = 9 + 1 = 10
\]
\[
BC^2 = (5-3)^2 + (5-6)^2 = 2^2 + (-1)^2 = 4 + 1 = 5
\]
(Not isosceles)
3. **For \( C(3, 4) \)**:
\[
AC^2 = (3-2)^2 + (4-4)^2 = 1^2 + 0^2 = 1 + 0 = 1
\]
\[
BC^2 = (3-3)^2 + (4-6)^2 = 0^2 + (-2)^2 = 0 + 4 = 4
\]
(Not isosceles)
4. **For \( C(0, 4) \)**:
\[
AC^2 = (0-2)^2 + (4-4)^2 = (-2)^2 + 0^2 = 4 + 0 = 4
\]
\[
BC^2 = (0-3)^2 + (4-6)^2 = (-3)^2 + (-2)^2 = 9 + 4 = 13
\]
(Not isosceles)
None of the options given would make \( \triangle ABC \) isosceles based on this analysis; however, the point \( C(3, 4) \) results in one distance being the same as another (the distance \( AC \) is 1 while \( AB \) is 5).
If none of the distances led to equality, we may need to rethink our options. However, the provided options do not include points that will satisfy all conditions for \( C \) making \( \triangle ABC \) isosceles.
If you have no issues with rounding decisions or interpretations of equal lengths, you may need different coordinates for \( C \). The values previously considered do not isolate or resolve any distances equitably and lead to an isosceles form hence necessitating explicit or additional coordinate ranges to exact the geometric outputs.
Let's denote the coordinates of the points:
- \( A(2, 4) \)
- \( B(3, 6) \)
- \( C(x, y) \) (where \( (x, y) \) will be one of the options provided)
We can compute the squared distances to avoid dealing with square roots:
1. **Distance from \( A \) to \( B \)**:
\[
AB^2 = (3-2)^2 + (6-4)^2 = 1^2 + 2^2 = 1 + 4 = 5
\]
Now, we will check each option for point \( C \) to see if it results in two equal distances:
1. **For \( C(5, 4) \)**:
\[
AC^2 = (5-2)^2 + (4-4)^2 = 3^2 + 0^2 = 9 + 0 = 9
\]
\[
BC^2 = (5-3)^2 + (4-6)^2 = 2^2 + (-2)^2 = 4 + 4 = 8
\]
(Not isosceles)
2. **For \( C(5, 5) \)**:
\[
AC^2 = (5-2)^2 + (5-4)^2 = 3^2 + 1^2 = 9 + 1 = 10
\]
\[
BC^2 = (5-3)^2 + (5-6)^2 = 2^2 + (-1)^2 = 4 + 1 = 5
\]
(Not isosceles)
3. **For \( C(3, 4) \)**:
\[
AC^2 = (3-2)^2 + (4-4)^2 = 1^2 + 0^2 = 1 + 0 = 1
\]
\[
BC^2 = (3-3)^2 + (4-6)^2 = 0^2 + (-2)^2 = 0 + 4 = 4
\]
(Not isosceles)
4. **For \( C(0, 4) \)**:
\[
AC^2 = (0-2)^2 + (4-4)^2 = (-2)^2 + 0^2 = 4 + 0 = 4
\]
\[
BC^2 = (0-3)^2 + (4-6)^2 = (-3)^2 + (-2)^2 = 9 + 4 = 13
\]
(Not isosceles)
None of the options given would make \( \triangle ABC \) isosceles based on this analysis; however, the point \( C(3, 4) \) results in one distance being the same as another (the distance \( AC \) is 1 while \( AB \) is 5).
If none of the distances led to equality, we may need to rethink our options. However, the provided options do not include points that will satisfy all conditions for \( C \) making \( \triangle ABC \) isosceles.
If you have no issues with rounding decisions or interpretations of equal lengths, you may need different coordinates for \( C \). The values previously considered do not isolate or resolve any distances equitably and lead to an isosceles form hence necessitating explicit or additional coordinate ranges to exact the geometric outputs.