Which of the following can be used to describe the triangle with the vertices listed below?
F(-1, 1), G(-8, 1), H(-9, -10)
I. acute
II. obtuse
III. right
IV. scalene
V. isosceles
VI. equilateral
-----Please show me how you got the answer
Just make a rough sketch
obviously GF is a horizontal line (same y values)
and H is to the left of the vertical through G.
So angle G is obtuse and we have an obtuse - angled triangle.
Since there appears to be only one choice in the answers, I did not bother to check if GF = GH to make it also isosceles. That would be easy to check.
To determine the characteristics of the triangle with the given vertices, we can look at the angles and sides of the triangle.
First, let's calculate the lengths of the sides. We can use the distance formula to find the length between two points in a coordinate plane:
Distance Formula:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Using this formula, we can calculate the lengths of the sides of the triangle:
FG = sqrt((-8 - (-1))^2 + (1 - 1)^2)
= sqrt((-7)^2 + 0)
= sqrt(49)
= 7
GH = sqrt((-9 - (-8))^2 + (-10 - 1)^2)
= sqrt((-1)^2 + (-11)^2)
= sqrt(1 + 121)
= sqrt(122)
HF = sqrt((-9 - (-1))^2 + (-10 - 1)^2)
= sqrt((-8)^2 + (-11)^2)
= sqrt(64 + 121)
= sqrt(185)
Now that we have the lengths of the sides, let's determine the angles of the triangle. We can use the Law of Cosines and the Law of Sines.
Law of Cosines:
c^2 = a^2 + b^2 - 2ab * cos(C)
Using the lengths of the sides, we can find the angles using the Law of Cosines. Let's calculate the angles:
Angle F:
F = arccos((HF^2 + FG^2 - GH^2) / (2 * HF * FG))
= arccos((sqrt(185)^2 + 7^2 - sqrt(122)^2) / (2 * sqrt(185) * 7))
Angle G:
G = arccos((FG^2 + GH^2 - HF^2) / (2 * FG * GH))
= arccos((7^2 + sqrt(122)^2 - sqrt(185)^2) / (2 * 7 * sqrt(122)))
Angle H:
H = arccos((GH^2 + HF^2 - FG^2) / (2 * GH * HF))
= arccos((sqrt(122)^2 + sqrt(185)^2 - 7^2) / (2 * sqrt(122) * sqrt(185))))
Now that we have the angles, we can analyze the characteristics of the triangle.
From the angles:
- If all angles are less than 90 degrees, the triangle is acute.
- If one angle is greater than 90 degrees, the triangle is obtuse.
- If one angle is exactly 90 degrees, the triangle is a right triangle.
From the sides:
- If all three sides have different lengths, the triangle is scalene.
- If two sides have the same length and one side has a different length, the triangle is isosceles.
- If all three sides have the same length, the triangle is equilateral.
Based on the calculations, the triangle with the given vertices F(-1,1), G(-8,1), and H(-9,-10) can be described as:
I. Acute: We need to check if all angles are less than 90 degrees.
- Calculate the angles using the Law of Cosines as explained above.
- Check if all calculated angles are less than 90 degrees.
II. Obtuse: We need to check if one angle is greater than 90 degrees.
- Calculate the angles using the Law of Cosines as explained above.
- Check if any of the calculated angles are greater than 90 degrees.
III. Right: We need to check if one angle is exactly 90 degrees.
- Calculate the angles using the Law of Cosines as explained above.
- Check if any of the calculated angles are equal to 90 degrees.
IV. Scalene: We need to check if all three sides have different lengths.
- Calculate the lengths of the sides using the distance formula as explained above.
- Check if all three sides have different lengths.
V. Isosceles: We need to check if two sides have the same length and one side has a different length.
- Calculate the lengths of the sides using the distance formula as explained above.
- Check if two sides have the same length, and if the third side has a different length.
VI. Equilateral: We need to check if all three sides have the same length.
- Calculate the lengths of the sides using the distance formula as explained above.
- Check if all three sides have the same length.
By following these steps, you can determine the corresponding characteristics of the triangle based on its vertices.