the coordinates of triangle KLM are
K(-2,-5), L(0,1), M(6,-2)
What type of triangle is KLM?
A) obtuse isosceles
B) acute scalene
C) right isosceles
D) right scalene
Find the length of each of the 3 sides using the formula you must have learned to find the length of a line if given the 2 endpoints.
Then knowing the definitions of each of the choices, pick the correct answer.
To determine the type of triangle KLM, we need to examine the angles and sides of the triangle.
1. Angles:
We can find the three angles of triangle KLM by using the coordinates of its vertices.
Angle KLM:
Using the coordinates of points K(-2,-5), L(0,1), and M(6,-2), we can find the length of the sides KL, LM, and MK. Then, we can use the Law of Cosines to find the angle KLM.
Length of side KL = √[(-2 - 0)^2 + (-5 - 1)^2]
= √[4 + 36]
= √40 = 2√10
Length of side LM = √[(0 - 6)^2 + (1 + 2)^2]
= √[36 + 9]
= √45 = 3√5
Length of side MK = √[(-2 - 6)^2 + (-5 + 2)^2]
= √[64 + 9]
= √73
Using the Law of Cosines:
cos(KLM) = (KL^2 + LM^2 - MK^2) / (2 * KL * LM)
= (4√10)^2 + (3√5)^2 - (√73)^2) / (2 * (4√10) * (3√5))
By solving this equation, we can find the value of cos(KLM).
2. Sides:
Next, we need to examine the lengths of the sides KL, LM, and MK.
- If all three sides are of different lengths, it is a scalene triangle.
- If two sides have the same length, it is an isosceles triangle.
- If all three sides have the same length, it is an equilateral triangle.
Now, based on the above calculations, we can determine the type of triangle KLM.
Solution:
Using the calculated values, we can find the following:
cos(KLM) ≈ 0.069
Since the value of cos(KLM) is positive and less than 1, the angle KLM is acute (less than 90 degrees). Therefore, triangle KLM is an acute triangle.
Looking at the lengths of the sides KL, LM, and MK, we can see that they are all different. Thus, triangle KLM is a scalene triangle.
Therefore, the answer is B) acute scalene