Find angle A of a triangle whose vertices are A (3,3) , B(-3,1) and C (-1,-3)
Or, you can take the slopes of the sides.
AB has slope 1/3
AC has slope 3/2
So, measuring as usual, angle A is
arctan(3/2)-arctan(1/3) = 236.31°-198.43° = 37.88°
sketch it first
then find the lengths of the sides
for example
c^2 = 2^2 + 6^2 = 40
c = 2 sqrt 10
then use law of cosines
a^2 = b^2 + c^2 - 2 b c cos A
Well, calculating angles can be a little tricky, but don't worry, I'm here to help! Let's find angle A together.
To find angle A, we need to use the concept of slopes. The slope of a line passing through two points (x₁, y₁) and (x₂, y₂) is given by the formula:
m = (y₂ - y₁) / (x₂ - x₁)
First, let's find the slopes of line AB and line AC.
The slope of line AB is:
m₁ = (1 - 3) / (-3 - 3)
= (-2) / (-6)
= 1/3
The slope of line AC is:
m₂ = (-3 - 3) / (-1 - 3)
= (-6) / (-4)
= 3/2
Now, we can use the slope formula to find the tangent of angle A, and then use the inverse tangent function to find the actual angle.
tanA = (m₂ - m₁) / (1 + (m₁ * m₂))
= (3/2 - 1/3) / (1 + (1/3 * 3/2))
= (5/6) / (1 + (1/6))
= (5/6) / (7/6)
= 5/7
Using the inverse tangent function (tan⁻¹), we can find angle A:
A = tan⁻¹(5/7)
≈ 0.62 radians
≈ 35.54 degrees
So, the angle A of the triangle is approximately 35.54 degrees. I hope this helps, and remember, triangles can be acute, obtuse, or even a little bit funny!
To find the angle A of a triangle, you can use the formula:
cos(A) = (b² + c² - a²) / (2 * b * c),
where a, b, and c are the lengths of the sides opposite angles A, B, and C respectively.
In this case, to find angle A, we need to find the lengths of sides b and c. We can do this by calculating the distance between the vertices of the triangle.
1. Calculate the length of side b (opposite angle B):
- Coordinates of points B(-3,1) and C(-1,-3)
- Using the distance formula, d = √((x2 - x1)² + (y2 - y1)²)
- Distance between B and C = √((-1 - -3)² + (-3 - 1)²)
- Distance between B and C = √((2)² + (-4)²)
- Distance between B and C = √(4 + 16) = √20
2. Calculate the length of side c (opposite angle C):
- Coordinates of points C(-1,-3) and A(3,3)
- Distance between C and A = √((3 - -1)² + (3 - -3)²)
- Distance between C and A = √((4)² + (6)²)
- Distance between C and A = √(16 + 36) = √52 = 2√13
3. Calculate the length of side a (opposite angle A):
- Using the distance formula, d = √((x2 - x1)² + (y2 - y1)²)
- Distance between A and B = √((-3 - 3)² + (1 - 3)²)
- Distance between A and B = √((-6)² + (-2)²)
- Distance between A and B = √(36 + 4) = √40 = 2√10
4. Using the lengths of sides b = √20 and c = 2√13, and length of opposite side a = 2√10, apply the formula:
cos(A) = (b² + c² - a²) / (2 * b * c)
cos(A) = (√20)² + (2√13)² - (2√10)² / (2 * √20 * 2√13)
cos(A) = 20 + 52 - 40 / (4√260)
cos(A) = 32 / (4√260) = 8 / √260 = 2 / √65
5. To find the value of angle A, take the inverse cosine of cos(A):
A = arccos(2 / √65)
Using a calculator, calculate the value of arccos(2 / √65) to get the final result for angle A in degrees or radians depending on the calculator settings.