Which of the following can be used to describe the triangle with the vertices listed below?
F(-1, 2), G(-10, 2), H(-12, -11)
I. acute
II. obtuse
III. right
IV. scalene
V. isosceles
VI. equilateral
--------Please help!!! Explain.
F and G lie on the same horizontal line.
H lies a bit to the left of G, and well below it.
That makes angle FGH obtuse.
If only one answer may be chosen, then (II)
FG=9
FH=17.02
GH=13.15
So, (IV) is also true
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To determine the description of the given triangle (acute, obtuse, right, scalene, isosceles, or equilateral), we need to analyze its angles and side lengths.
First, let's plot the triangle with the given vertices:
Triangle FGH:
F(-1, 2)
G(-10, 2)
H(-12, -11)
To analyze the angles, we can use the concept of slope. The slope formula is:
m = (y2 - y1) / (x2 - x1)
Let's calculate the slopes of the sides FG, GH, and FH:
1. Slope of FG: m = (2 - 2) / (-10 - (-1)) = 0 / -9 = 0
2. Slope of GH: m = (2 - (-11)) / (-10 - (-12)) = 13 / 2.
Based on the slopes, we can conclude:
1. The slope of FG is 0, indicating a horizontal line. Therefore, angle F is a right angle.
2. The slope of GH is positive (13/2), indicating an upward sloping line. So, angle G is acute.
To determine the third angle, we can use the fact that the sum of angles in a triangle is always 180 degrees. Since we know angle F is a right angle (90 degrees) and angle G is acute, angle H must be obtuse.
Now, let's analyze the side lengths:
1. Length of FG: Distance Formula = square root(((-10) - (-1))^2 + (2 - 2)^2) = 9 units.
2. Length of GH: Distance Formula = square root(((-10) - (-12))^2 + (2 - (-11))^2) = square root(4 + 195) ≈ 14.81 units.
3. Length of FH: Distance Formula = square root(((-12) - (-1))^2 + ((-11) - 2)^2) = square root(121 + 169) ≈ 17.69 units.
Based on the side lengths, we can conclude:
1. All sides (FG, GH, FH) have different lengths, so the triangle is scalene.
2. Since none of the sides are equal, the triangle is not isosceles or equilateral.
In summary, the triangle with vertices F(-1, 2), G(-10, 2), and H(-12, -11) can be described as:
- An acute triangle (angle G is acute).
- A right triangle (angle F is a right angle).
- An obtuse triangle (angle H is obtuse).
- A scalene triangle (all sides have different lengths).
- It is not isosceles or equilateral.