the triangle formed by points L(-1, 7), M(1, -2), and N(6, 1) is an acute scalene triangle with side lengths of approximately 7.2, 7.2, and 8.2, and angles of approximately 83, 47, and 50 degrees.
The triangle formed by points L(-1, 7), M(1, -2), and N(6, 1) is an acute scalene triangle.
Acute: All three angles of the triangle are less than 90 degrees. You can measure this with a protractor on the image, or use the fact that the sum of the angles in a triangle is 180 degrees, and none of the angles here are close to 90 degrees.
Scalene: All three sides of the triangle have different lengths. As you can see from the image, no two sides have the same length.
To find the lengths of the sides and the measures of the angles, we can use the distance formula and trigonometry.
Side lengths:
LM = √((1 - (-1))^2 + (-2 - 7)^2) = √(52) ≈ 7.2
MN = √((6 - 1)^2 + (1 - (-2))^2) = √(52) ≈ 7.2
NL = √((6 - (-1))^2 + (1 - 7)^2) = √(68) ≈ 8.2
Angle measures:
We can use the Law of Cosines to find the measures of the angles. For example, to find the measure of angle LMN, we can use the formula: cos(LMN) = (MN^2 + LM^2 - NL^2) / (2 * MN * LM) Plugging in the values we calculated for the side lengths, we get cos(LMN) ≈ 0.174. Therefore, LMN ≈ 83 degrees (using the inverse cosine function).
Similarly, you can find the other angles using the Law of Cosines or the Law of Sines.
In conclusion, the triangle formed by points L(-1, 7), M(1, -2), and N(6, 1) is an acute scalene triangle with side lengths of approximately 7.2, 7.2, and 8.2, and angles of approximately 83, 47, and 50 degrees.