Let's solve the problem step-by-step for [tex]$\triangle ABC$[/tex] with vertices [tex]$A (-2,5)$[/tex], [tex]$B(-4,-2)$[/tex], and [tex]$C(3,-4)$[/tex].
### Step 1: Calculate the length of [tex]\( AB \)[/tex]
Using the distance formula:
[tex]\[ AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
For points [tex]\( A (-2,5) \)[/tex] and [tex]\( B(-4,-2) \)[/tex]:
[tex]\[ AB = \sqrt{((-4) - (-2))^2 + ((-2) - 5)^2} = \sqrt{(-2)^2 + (-7)^2} = \sqrt{4 + 49} = \sqrt{53} \approx 7.28 \][/tex]
So, the length of [tex]\( AB \)[/tex] is [tex]\( 7.28 \)[/tex].
### Step 2: Calculate the length of [tex]\( AC \)[/tex]
Using the distance formula:
[tex]\[ AC = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
For points [tex]\( A (-2,5) \)[/tex] and [tex]\( C(3,-4) \)[/tex]:
[tex]\[ AC = \sqrt{(3 - (-2))^2 + (-4 - 5)^2} = \sqrt{(3 + 2)^2 + (-4 - 5)^2} = \sqrt{5^2 + (-9)^2} = \sqrt{25 + 81} = \sqrt{106} \approx 10.30 \][/tex]
So, the length of [tex]\( AC \)[/tex] is [tex]\( 10.30 \)[/tex].
### Step 3: Calculate the length of [tex]\( BC \)[/tex]
Using the distance formula:
[tex]\[ BC = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
For points [tex]\( B (-4,-2) \)[/tex] and [tex]\( C(3,-4) \)[/tex]:
[tex]\[ BC = \sqrt{(3 - (-4))^2 + (-4 - (-2))^2} = \sqrt{(3 + 4)^2 + (-4 + 2)^2} = \sqrt{7^2 + (-2)^2} = \sqrt{49 + 4} = \sqrt{53} \approx 7.28 \][/tex]
So, the length of [tex]\( BC \)[/tex] is [tex]\( 7.28 \)[/tex].
### Step 4: Determine the type of triangle
We compare the lengths of [tex]\( AB \)[/tex], [tex]\( AC \)[/tex], and [tex]\( BC \)[/tex]:
- [tex]\( AB \approx 7.28 \)[/tex]
- [tex]\( AC \approx 10.30 \)[/tex]
- [tex]\( BC \approx 7.28 \)[/tex]
Since two sides ([tex]\( AB \)[/tex] and [tex]\( BC \)[/tex]) are equal in length, [tex]\( \triangle ABC \)[/tex] is an isosceles triangle.
### Conclusion
Based on the calculations:
- The length of [tex]\( AB \)[/tex] is [tex]\( 7.28 \)[/tex]
- The length of [tex]\( AC \)[/tex] is [tex]\( 10.30 \)[/tex]
- The length of [tex]\( BC \)[/tex] is [tex]\( 7.28 \)[/tex]
- Therefore, the triangle is isosceles
Fill in the dropdowns as follows:
- The length of [tex]\( AB \)[/tex] is 7.28
- The length of [tex]\( AC \)[/tex] is 10.30
- The length of [tex]\( BC \)[/tex] is 7.28
- Therefore, the triangle is isosceles
