Triangle KLM has vertices K (3,2) L(-1,5) and M(-3,-7) write the angles in order from the least to greatest measure show all work
To find the angles of triangle KLM, we can use the distance formula and the law of cosines. Here are the steps:
Step 1: Find the lengths of the sides of triangle KLM using the distance formula. The distance formula states that the distance between two points (x1, y1) and (x2, y2) is given by:
d = √((x2 - x1)^2 + (y2 - y1)^2)
- Length KL:
KL = √((-1 - 3)^2 + (5 - 2)^2)
= √((-4)^2 + (3)^2)
= √(16 + 9)
= √25
= 5
- Length KM:
KM = √((-3 - 3)^2 + (-7 - 2)^2)
= √((-6)^2 + (-9)^2)
= √(36 + 81)
= √117
≈ 10.82
- Length LM:
LM = √((-3 - (-1))^2 + (-7 - 5)^2)
= √((-3 + 1)^2 + (-7 - 5)^2)
= √((-2)^2 + (-12)^2)
= √(4 + 144)
= √148
≈ 12.17
Step 2: Use the law of cosines to find the measures of the angles.
- Angle K:
By the law of cosines, we have:
cos(K) = (KL^2 + KM^2 - LM^2) / (2 * KL * KM)
cos(K) = (5^2 + 10.82^2 - 12.17^2) / (2 * 5 * 10.82)
cos(K) = (25 + 117.0724 - 148.2489) / (108.2)
cos(K) = -0.3796
Now, use the inverse cosine function to find the measure of angle K:
K = cos^(-1)(-0.3796)
K ≈ 111.16 degrees
- Angle L:
Using the law of cosines again, we have:
cos(L) = (KL^2 + LM^2 - KM^2) / (2 * KL * LM)
cos(L) = (5^2 + 12.17^2 - 10.82^2) / (2 * 5 * 12.17)
cos(L) = (25 + 148.6489 - 117.0724) / (121.7)
cos(L) = 0.2112
L = cos^(-1)(0.2112)
L ≈ 77.22 degrees
- Angle M:
Since the sum of the angles in a triangle is always 180 degrees, we can find angle M by subtracting angles K and L from 180 degrees:
M = 180 - K - L
M = 180 - 111.16 - 77.22
M ≈ -8.38 degrees
Note: The negative value for angle M indicates that M is an obtuse angle.
Therefore, the angles of triangle KLM from least to greatest measure are approximately:
L ≈ 77.22 degrees
K ≈ 111.16 degrees
M ≈ -8.38 degrees
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