This question is suppose to be simple but i keep getting it wrong
Triangle LMN has vertices L(1, 6), M(6, 3) and N(5, 7). What type of triangle is Triangle LMN? Select all that apply
Scalene triangle
Right triangle
Isosceles triangle
Equilateral triangle
Obtuse triangle
Acute triangle
I got Isosceles Triange and acute but i keep getting it worong
LM = √(5^2+3^2) = √34
LN = √(4^2+1^2) = √17
MN = √(1^1+4^2) = √17
So, it is isosceles.
But since
√17^2 + √17^2 = √34^2
17+17 = 34
it is a right triangle.
How do you know its a right angle when the answer is 34 i thought it has to be 90?
I'm not talking about angles, but sides. Remember the Pythagorean Theorem?
a^2 + b^2 = c^2
To determine the type of triangle, we need to analyze the side lengths and angles of Triangle LMN. Here's a step-by-step guide to finding the correct answers:
1. Calculate the side lengths:
- Side LM: Distance formula: √((x2 - x1)^2 + (y2 - y1)^2)
LM = √((6 - 1)^2 + (3 - 6)^2) = √(25 + 9) = √34
- Side MN: Distance formula:
MN = √((5 - 6)^2 + (7 - 3)^2) = √(1 + 16) = √17
- Side NL: Distance formula:
NL = √((1 - 5)^2 + (6 - 7)^2) = √(16 + 1) = √17
2. Compare the side lengths:
- If all sides are different lengths, it is a Scalene triangle.
- If two sides are equal but the third side is different, it is an Isosceles triangle.
- If all three sides are equal, it is an Equilateral triangle.
Based on the side lengths LM = √34, MN = √17, and NL = √17, we can confirm that Triangle LMN is a Scalene triangle.
3. Calculate the angles:
- Angle L: Use the Law of Cosines: cos(L) = (b^2 + c^2 - a^2) / (2bc), where a, b, and c are the side lengths.
cos(L) = (17 + 17 - 34) / (2 * √17 * √17) = 0
L = arccos(0) = 90 degrees
- Angle M: Use the Law of Cosines:
cos(M) = (34 + 17 - 17) / (2 * √34 * √17) = 1 / √2
M = arccos(1 / √2) ≈ 45 degrees
- Angle N: Use the Law of Cosines:
cos(N) = (34 + 34 - 17) / (2 * √34 * √34) = 1 / 2
N = arccos(1 / 2) ≈ 60 degrees
4. Analyze the angles:
- If one angle measures exactly 90 degrees, it is a Right triangle.
- If all angles are less than 90 degrees, it is an Acute triangle.
- If one angle measures more than 90 degrees, it is an Obtuse triangle.
Based on the angles L = 90 degrees, M ≈ 45 degrees, and N ≈ 60 degrees, we can conclude that Triangle LMN is a Right triangle and an Acute triangle.
In summary, Triangle LMN is a Scalene triangle, a Right triangle, and an Acute triangle.