Answer :
To address this question regarding the oblique triangle, let's proceed step-by-step:
### a) Label the vertices of the triangle, and list them below.
Given a triangle with vertices [tex]\(A\)[/tex], [tex]\(B\)[/tex], and [tex]\(C\)[/tex], we denote their coordinates as:
- [tex]\(A = (x_1, y_1)\)[/tex]
- [tex]\(B = (x_2, y_2)\)[/tex]
- [tex]\(C = (x_3, y_3)\)[/tex]
### b) Use the distance formula to determine the length of each side.
The distance formula is:
[tex]\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
Let’s calculate the lengths of each side:
1. Length of [tex]\( AB \)[/tex]:
[tex]\[ AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
2. Length of [tex]\( BC \)[/tex]:
[tex]\[ BC = \sqrt{(x_3 - x_2)^2 + (y_3 - y_2)^2} \][/tex]
3. Length of [tex]\( CA \)[/tex]:
[tex]\[ CA = \sqrt{(x_3 - x_1)^2 + (y_3 - y_1)^2} \][/tex]
### c) Use the cosine law to determine the largest angle.
The cosine law (or law of cosines) is usually stated as:
[tex]\[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \][/tex]
Where [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] are the lengths of the sides opposite the angles [tex]\( A \)[/tex], [tex]\( B \)[/tex], and [tex]\( C \)[/tex] respectively.
First, identify the longest side among [tex]\( AB \)[/tex], [tex]\( BC \)[/tex], and [tex]\( CA \)[/tex]. The largest angle is opposite the longest side. Let's assume [tex]\( CA \)[/tex] is the longest side; hence the largest angle is [tex]\( \angle B \)[/tex].
Using the cosine law to find [tex]\( \angle B \)[/tex]:
[tex]\[ CA^2 = AB^2 + BC^2 - 2 \cdot AB \cdot BC \cdot \cos(B) \][/tex]
Solving for [tex]\( \cos(B) \)[/tex]:
[tex]\[ \cos(B) = \frac{AB^2 + BC^2 - CA^2}{2 \cdot AB \cdot BC} \][/tex]
Then, use the inverse cosine function ([tex]\( \cos^{-1} \)[/tex]) to find [tex]\( \angle B \)[/tex]:
[tex]\[ B = \cos^{-1}\left( \frac{AB^2 + BC^2 - CA^2}{2 \cdot AB \cdot BC} \right) \][/tex]
Convert the angle from radians to degrees if necessary.
### d) Use the sine law to determine one other angle.
The sine law (or law of sines) states:
[tex]\[ \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} \][/tex]
Suppose you want to find angle [tex]\( A \)[/tex]. We can use the following proportion assuming we already found angle [tex]\( B \)[/tex]:
[tex]\[ \frac{AB}{\sin(A)} = \frac{CA}{\sin(B)} \][/tex]
To find [tex]\( \sin(A) \)[/tex]:
[tex]\[ \sin(A) = \frac{AB \cdot \sin(B)}{CA} \][/tex]
Convert angle [tex]\( B \)[/tex] to degrees if it isn’t already, and then find [tex]\( \angle A \)[/tex]:
[tex]\[ A = \sin^{-1}\left( \frac{AB \cdot \sin(B)}{CA} \right) \][/tex]
### Summary
1. Vertices: A = [tex]\((x_1, y_1)\)[/tex], B = [tex]\((x_2, y_2)\)[/tex], C = [tex]\((x_3, y_3)\)[/tex]
2. Side Lengths:
- [tex]\( AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)[/tex]
- [tex]\( BC = \sqrt{(x_3 - x_2)^2 + (y_3 - y_2)^2} \)[/tex]
- [tex]\( CA = \sqrt{(x_3 - x_1)^2 + (y_3 - y_1)^2} \)[/tex]
3. Largest Angle: Using cosine law on the largest side.
4. Another Angle: Using sine law with the determined largest angle.
By following these steps carefully, you can determine the lengths of the sides, the largest angle, and another angle in the triangle.
### a) Label the vertices of the triangle, and list them below.
Given a triangle with vertices [tex]\(A\)[/tex], [tex]\(B\)[/tex], and [tex]\(C\)[/tex], we denote their coordinates as:
- [tex]\(A = (x_1, y_1)\)[/tex]
- [tex]\(B = (x_2, y_2)\)[/tex]
- [tex]\(C = (x_3, y_3)\)[/tex]
### b) Use the distance formula to determine the length of each side.
The distance formula is:
[tex]\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
Let’s calculate the lengths of each side:
1. Length of [tex]\( AB \)[/tex]:
[tex]\[ AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
2. Length of [tex]\( BC \)[/tex]:
[tex]\[ BC = \sqrt{(x_3 - x_2)^2 + (y_3 - y_2)^2} \][/tex]
3. Length of [tex]\( CA \)[/tex]:
[tex]\[ CA = \sqrt{(x_3 - x_1)^2 + (y_3 - y_1)^2} \][/tex]
### c) Use the cosine law to determine the largest angle.
The cosine law (or law of cosines) is usually stated as:
[tex]\[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \][/tex]
Where [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] are the lengths of the sides opposite the angles [tex]\( A \)[/tex], [tex]\( B \)[/tex], and [tex]\( C \)[/tex] respectively.
First, identify the longest side among [tex]\( AB \)[/tex], [tex]\( BC \)[/tex], and [tex]\( CA \)[/tex]. The largest angle is opposite the longest side. Let's assume [tex]\( CA \)[/tex] is the longest side; hence the largest angle is [tex]\( \angle B \)[/tex].
Using the cosine law to find [tex]\( \angle B \)[/tex]:
[tex]\[ CA^2 = AB^2 + BC^2 - 2 \cdot AB \cdot BC \cdot \cos(B) \][/tex]
Solving for [tex]\( \cos(B) \)[/tex]:
[tex]\[ \cos(B) = \frac{AB^2 + BC^2 - CA^2}{2 \cdot AB \cdot BC} \][/tex]
Then, use the inverse cosine function ([tex]\( \cos^{-1} \)[/tex]) to find [tex]\( \angle B \)[/tex]:
[tex]\[ B = \cos^{-1}\left( \frac{AB^2 + BC^2 - CA^2}{2 \cdot AB \cdot BC} \right) \][/tex]
Convert the angle from radians to degrees if necessary.
### d) Use the sine law to determine one other angle.
The sine law (or law of sines) states:
[tex]\[ \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} \][/tex]
Suppose you want to find angle [tex]\( A \)[/tex]. We can use the following proportion assuming we already found angle [tex]\( B \)[/tex]:
[tex]\[ \frac{AB}{\sin(A)} = \frac{CA}{\sin(B)} \][/tex]
To find [tex]\( \sin(A) \)[/tex]:
[tex]\[ \sin(A) = \frac{AB \cdot \sin(B)}{CA} \][/tex]
Convert angle [tex]\( B \)[/tex] to degrees if it isn’t already, and then find [tex]\( \angle A \)[/tex]:
[tex]\[ A = \sin^{-1}\left( \frac{AB \cdot \sin(B)}{CA} \right) \][/tex]
### Summary
1. Vertices: A = [tex]\((x_1, y_1)\)[/tex], B = [tex]\((x_2, y_2)\)[/tex], C = [tex]\((x_3, y_3)\)[/tex]
2. Side Lengths:
- [tex]\( AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)[/tex]
- [tex]\( BC = \sqrt{(x_3 - x_2)^2 + (y_3 - y_2)^2} \)[/tex]
- [tex]\( CA = \sqrt{(x_3 - x_1)^2 + (y_3 - y_1)^2} \)[/tex]
3. Largest Angle: Using cosine law on the largest side.
4. Another Angle: Using sine law with the determined largest angle.
By following these steps carefully, you can determine the lengths of the sides, the largest angle, and another angle in the triangle.
