Question
Point A
𝐴
lies at (1,2)
(
1
,
2
)
and point B
𝐵
lies at (4,5)
(
4
,
5
)
. Which point in needed so that △ABC
△
𝐴
𝐵
𝐶
is isosceles? (1 point)
Responses
C (3,2)
𝐶
(
3
,
2
)
C (3,2)
𝐶
(
3
,
2
)
C (7,8)
𝐶
(
7
,
8
)
C (7,8)
𝐶
(
7
,
8
)
C (1,5)
𝐶
(
1
,
5
)
C (1,5)
𝐶
(
1
,
5
)
C (4,2)
𝐴
lies at (1,2)
(
1
,
2
)
and point B
𝐵
lies at (4,5)
(
4
,
5
)
. Which point in needed so that △ABC
△
𝐴
𝐵
𝐶
is isosceles? (1 point)
Responses
C (3,2)
𝐶
(
3
,
2
)
C (3,2)
𝐶
(
3
,
2
)
C (7,8)
𝐶
(
7
,
8
)
C (7,8)
𝐶
(
7
,
8
)
C (1,5)
𝐶
(
1
,
5
)
C (1,5)
𝐶
(
1
,
5
)
C (4,2)
Answers
GPT-4o mini
To determine which point \( C \) makes triangle \( ABC \) isosceles, we need to find a point \( C \) such that at least two sides of triangle \( ABC \) have equal lengths.
We can calculate the lengths of the segments \( AC \), \( BC \), and \( AB \) for each of the candidate points \( C \):
1. **For C (3, 2)**:
- Length \( AC \):
\[
AC = \sqrt{(3 - 1)^2 + (2 - 2)^2} = \sqrt{2^2} = 2
\]
- Length \( BC \):
\[
BC = \sqrt{(3 - 4)^2 + (2 - 5)^2} = \sqrt{(-1)^2 + (-3)^2} = \sqrt{1 + 9} = \sqrt{10}
\]
- Length \( AB \):
\[
AB = \sqrt{(4 - 1)^2 + (5 - 2)^2} = \sqrt{3^2 + 3^2} = \sqrt{9 + 9} = \sqrt{18}
\]
2. **For C (7, 8)**:
- Length \( AC \):
\[
AC = \sqrt{(7 - 1)^2 + (8 - 2)^2} = \sqrt{6^2 + 6^2} = \sqrt{36 + 36} = \sqrt{72}
\]
- Length \( BC \):
\[
BC = \sqrt{(7 - 4)^2 + (8 - 5)^2} = \sqrt{3^2 + 3^2} = \sqrt{9 + 9} = \sqrt{18}
\]
- Length \( AB \):
\[
AB = \sqrt{18} \quad (\text{same as before})
\]
3. **For C (1, 5)**:
- Length \( AC \):
\[
AC = \sqrt{(1 - 1)^2 + (5 - 2)^2} = \sqrt{0 + 3^2} = 3
\]
- Length \( BC \):
\[
BC = \sqrt{(1 - 4)^2 + (5 - 5)^2} = \sqrt{(-3)^2 + 0^2} = 3
\]
- Length \( AB \):
\[
AB = \sqrt{18} \quad (\text{same as before})
\]
4. **For C (4, 2)**:
- Length \( AC \):
\[
AC = \sqrt{(4 - 1)^2 + (2 - 2)^2} = \sqrt{3^2 + 0^2} = 3
\]
- Length \( BC \):
\[
BC = \sqrt{(4 - 4)^2 + (2 - 5)^2} = \sqrt{0 + (-3)^2} = 3
\]
- Length \( AB \):
\[
AB = \sqrt{18} \quad (\text{same as before})
\]
From this analysis:
- For point C (3, 2), lengths \( AC \), \( BC \), and \( AB \) are not equal.
- For point C (7, 8), lengths \( BC \) and \( AB\) are equal \( (\sqrt{18}) \).
- For point C (1, 5), lengths \( AC \) and \( BC\) are equal (both are 3).
- For point C (4, 2), lengths \( AC \) and \( BC \) also are equal (both are 3).
Thus, both points \( C(1, 5) \) and \( C(4, 2) \) form an isosceles triangle \( ABC \).
**Answer:** C (1, 5) and C (4, 2) both make triangle \( ABC \) isosceles.
We can calculate the lengths of the segments \( AC \), \( BC \), and \( AB \) for each of the candidate points \( C \):
1. **For C (3, 2)**:
- Length \( AC \):
\[
AC = \sqrt{(3 - 1)^2 + (2 - 2)^2} = \sqrt{2^2} = 2
\]
- Length \( BC \):
\[
BC = \sqrt{(3 - 4)^2 + (2 - 5)^2} = \sqrt{(-1)^2 + (-3)^2} = \sqrt{1 + 9} = \sqrt{10}
\]
- Length \( AB \):
\[
AB = \sqrt{(4 - 1)^2 + (5 - 2)^2} = \sqrt{3^2 + 3^2} = \sqrt{9 + 9} = \sqrt{18}
\]
2. **For C (7, 8)**:
- Length \( AC \):
\[
AC = \sqrt{(7 - 1)^2 + (8 - 2)^2} = \sqrt{6^2 + 6^2} = \sqrt{36 + 36} = \sqrt{72}
\]
- Length \( BC \):
\[
BC = \sqrt{(7 - 4)^2 + (8 - 5)^2} = \sqrt{3^2 + 3^2} = \sqrt{9 + 9} = \sqrt{18}
\]
- Length \( AB \):
\[
AB = \sqrt{18} \quad (\text{same as before})
\]
3. **For C (1, 5)**:
- Length \( AC \):
\[
AC = \sqrt{(1 - 1)^2 + (5 - 2)^2} = \sqrt{0 + 3^2} = 3
\]
- Length \( BC \):
\[
BC = \sqrt{(1 - 4)^2 + (5 - 5)^2} = \sqrt{(-3)^2 + 0^2} = 3
\]
- Length \( AB \):
\[
AB = \sqrt{18} \quad (\text{same as before})
\]
4. **For C (4, 2)**:
- Length \( AC \):
\[
AC = \sqrt{(4 - 1)^2 + (2 - 2)^2} = \sqrt{3^2 + 0^2} = 3
\]
- Length \( BC \):
\[
BC = \sqrt{(4 - 4)^2 + (2 - 5)^2} = \sqrt{0 + (-3)^2} = 3
\]
- Length \( AB \):
\[
AB = \sqrt{18} \quad (\text{same as before})
\]
From this analysis:
- For point C (3, 2), lengths \( AC \), \( BC \), and \( AB \) are not equal.
- For point C (7, 8), lengths \( BC \) and \( AB\) are equal \( (\sqrt{18}) \).
- For point C (1, 5), lengths \( AC \) and \( BC\) are equal (both are 3).
- For point C (4, 2), lengths \( AC \) and \( BC \) also are equal (both are 3).
Thus, both points \( C(1, 5) \) and \( C(4, 2) \) form an isosceles triangle \( ABC \).
**Answer:** C (1, 5) and C (4, 2) both make triangle \( ABC \) isosceles.