To determine the possible coordinates of the vertex
A in an isosceles triangle ABC with base endpoints
B(1,3) and
C(−2,7), we need to find the coordinates that satisfy the property that
AB=AC.
Let's calculate the distance between
B(1,3) and potential coordinates of
A, as well as between
C(−2,7) and potential coordinates of
A.
1. Option A:
A(1,6)‌AB=√(1−1)2+(6−3)2=√0+9=√9=3
‌AC=√(1−(−2))2+(6−7)2=√32+(−1)2=√9+1=√10
Since
AB≠AC, Option
A is incorrect.
2. Option B :
A(−‌,5)Since
AB≠AC, Option B is incorrect.
3. Option C:
A(‌,6)Since
AB=AC, Option C is correct.
4. Option D:
A(−7,‌)Since
AB=AC, Option D is also correct.
Therefore, the correct coordinates of
A can be Option C:
A(‌,6) or Option D:
A(−7,‌).