Question

Point A
lies at (2,4)
and point B
lies at (3,6)
. Which point is needed so that △ABC
is isosceles?

Answer 1

is (5,4) a possible solution?

Answer 2

Is (5,5) a valid solution?

Answer 3

To form an isosceles triangle \( \triangle ABC \), we need to find a point \( C \) such that at least two sides of the triangle \( AB \), \( BC \), or \( AC \) are equal in length.

### Step 1: Calculate the length of side \( AB \)

Using the distance formula, the distance \( AB \) between points \( A(2, 4) \) and \( B(3, 6) \) is given by:

\[
AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{(3 - 2)^2 + (6 - 4)^2} = \sqrt{(1)^2 + (2)^2} = \sqrt{1 + 4} = \sqrt{5}
\]

### Step 2: Position of point \( C \)

To form an isosceles triangle, \( C \) can be placed in various positions such that either \( AC = BC \), or \( AC = AB \), or \( BC = AB \).

### Case 1: \( AC = AB \)

Point \( C \) must be at a distance of \( \sqrt{5} \) from point \( A(2, 4) \). The equation of the circle centered at \( A \) with radius \( \sqrt{5} \) is:

\[
(x - 2)^2 + (y - 4)^2 = 5
\]

### Case 2: \( BC = AB \)

Similarly, point \( C \) must be at a distance of \( \sqrt{5} \) from point \( B(3, 6) \). The equation for point \( C \) to lie this distance from \( B \) is:

\[
(x - 3)^2 + (y - 6)^2 = 5
\]

### Finding points \( C \)

You can find points \( C \) on either circle to form an isosceles triangle.

**Example Points:**
1. For \( AC = AB \):
- Let \( C \) be \( (2 + \sqrt{5}, 4) \) and \( (2 - \sqrt{5}, 4) \)
- Let \( C \) also be \( (2, 4 + \sqrt{5}) \) and \( (2, 4 - \sqrt{5}) \)

2. For \( BC = AB \):
- Let \( C \) be \( (3 + \sqrt{5}, 6) \) and \( (3 - \sqrt{5}, 6) \)
- Let \( C \) also be \( (3, 6 + \sqrt{5}) \) and \( (3, 6 - \sqrt{5}) \)

By exploring these formulas, you'll find multiple valid \( C \) points that will successfully create isosceles triangles.

Answer 4

To determine if point \( C(5, 4) \) can create an isosceles triangle \( \triangle ABC \) with points \( A(2, 4) \) and \( B(3, 6) \), we need to check if any two sides are equal.

### Step 1: Calculate the lengths of the sides

1. **Length of \( AC \)**:
\[
AC = \sqrt{(5 - 2)^2 + (4 - 4)^2} = \sqrt{(3)^2 + (0)^2} = \sqrt{9} = 3
\]

2. **Length of \( BC \)**:
\[
BC = \sqrt{(5 - 3)^2 + (4 - 6)^2} = \sqrt{(2)^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}
\]

3. **Length of \( AB \)** (previously calculated):
\[
AB = \sqrt{(3 - 2)^2 + (6 - 4)^2} = \sqrt{(1)^2 + (2)^2} = \sqrt{1 + 4} = \sqrt{5}
\]

### Step 2: Compare the lengths

Now we compare the lengths we found:

- \( AC = 3 \)
- \( BC = 2\sqrt{2} \approx 2.83 \)
- \( AB = \sqrt{5} \approx 2.24 \)

### Conclusion

Since none of these lengths are equal—\( AC \neq BC \), \( AC \neq AB \), and \( BC \neq AB \)—the triangle \( \triangle ABC \) with point \( C(5, 4) \) is **not isosceles**. Thus, while point \( (5, 4) \) is a valid point in space, it does not satisfy the condition for the triangle to be isosceles with respect to the given points \( A \) and \( B \).

Answer 5

To determine if point \( C(5, 5) \) creates an isosceles triangle \( \triangle ABC \) with points \( A(2, 4) \) and \( B(3, 6) \), we need to calculate the lengths of the sides \( AC \), \( BC \), and \( AB \).

### Step 1: Calculate the lengths of the sides

1. **Length of \( AC \)**:
\[
AC = \sqrt{(5 - 2)^2 + (5 - 4)^2} = \sqrt{(3)^2 + (1)^2} = \sqrt{9 + 1} = \sqrt{10}
\]

2. **Length of \( BC \)**:
\[
BC = \sqrt{(5 - 3)^2 + (5 - 6)^2} = \sqrt{(2)^2 + (-1)^2} = \sqrt{4 + 1} = \sqrt{5}
\]

3. **Length of \( AB \)** (already calculated):
\[
AB = \sqrt{(3 - 2)^2 + (6 - 4)^2} = \sqrt{(1)^2 + (2)^2} = \sqrt{1 + 4} = \sqrt{5}
\]

### Step 2: Compare the lengths

Now we compare the calculated lengths:

- \( AC = \sqrt{10} \)
- \( BC = \sqrt{5} \)
- \( AB = \sqrt{5} \)

### Conclusion

Here we see that \( BC = AB = \sqrt{5} \). Since at least two sides are equal, \( \triangle ABC \) with \( C(5, 5) \) is indeed an **isosceles triangle**. Thus, \( (5, 5) \) is a valid solution for creating an isosceles triangle with points \( A(2, 4) \) and \( B(3, 6) \).