Triangle LM has vertices K(3,2) L(-1,5) and M(-3,-7) Write the angles in order from the least to greatest measure
find the side lengths.
By the law of since, the angle measures are in the same order as the side lengths.
To find the angles of a triangle given the coordinates of its vertices, we can use the distance formula and the Pythagorean theorem. First, calculate the lengths of the sides of the triangle:
Side KL:
The distance formula is given by:
d = √[(x₂ - x₁)² + (y₂ - y₁)²]
d(KL) = √[(-1 - 3)² + (5 - 2)²]
= √[(-4)² + (3)²]
= √[16 + 9]
= √25
= 5
Side KM:
d(KM) = √[(-3 - 3)² + (-7 - 2)²]
= √[(-6)² + (-9)²]
= √[36 + 81]
= √117
Side LM:
d(LM) = √[(-3 - (-1))² + (-7 - 5)²]
= √[(-2)² + (-12)²]
= √[4 + 144]
= √148
= 2√37
Next, use the Law of Cosines to find the measures of the angles:
Cosine Rule:
c² = a² + b² - 2ab * Cos(C)
Angle K:
Cos(K) = (KL² + KM² - LM²) / 2(KL)(KM)
= (5² + (√117)² - (2√37)²) / (2)(5)(√117)
= (25 + 117 - 148) / (10√117)
= (142 - 148) / (10√117)
= -6 / 10√117
Angle L:
Cos(L) = (LM² + KL² - KM²) / 2(LM)(KL)
= ((2√37)² + 5² - (√117)²) / (2)(2√37)(5)
= (4(37) + 25 - 117) / (20√37)
= (148 + 25 - 117) / (20√37)
= 56 / (20√37)
= 14 / 5√37
Angle M:
Angle M is found by subtracting the sum of Angles K and L from 180°. Since we now know the measures of Angles K and L, we can substitute them in.
Angle M = 180° - Angle K - Angle L
= 180° - arccos(-6 / 10√117) - arccos(14 / 5√37)
To find these angles, you can substitute the values into a scientific calculator or use a trigonometric identity table.
Method 1:
use the slope of the 3 lines to find the angles the lines make with the x-axis.
Then subtract the angles to determine the angle between the lines
method 2:
use the formula tanØ = |(m1-m2)/(1+m1m2)|
where Ø is the angle between lines with slope m1 and m2
(this is really the same as method2)
method 3:
find the lengths of each line, then use the cosine law to find the angle between one pair to find one angle, then the sine law to find a second angle, then the sum of the 3 angles of a triangle to find the 3rd angle
I will find the smallest angle using all 3 methods
let Ø be the angle between LM and KM
method1:
slope of LM = 12/2 = 6, so LM makes an angle of 80.5° with the x-axis
slope of KM = 3/2 , so KM makes an angle of 56.3° with the x-axis
Ø = 80.5 - 56.3 = 24.2°
.... repeat for the other 2 angles
Method 2
tanØ = |(6-1.5)/(1 + 6(1.5)|
= 4.5/10 = 9/20
Ø = arctan(9/20) = 24.2°
etc
method 3
using distance formulas:
LM = √148 , LK = 5 , KM = √117
5^2 = √148^2 + √117^2 - 2√148√117cosØ
cosØ = (148 + 117 - 25)/(2√117√148) = .91192..
Ø = arccos(.9112..) = 24.2°
etc