The vertices of a triangle are listed below.
H(-2,4), I(22,4), J(10,-1)
Which of the following correctly classifies the triangle?
1.The triangle is an obtuse isosceles triangle.
2.The triangle is a right scalene triangle.
3.The triangle is an obtuse scalene triangle.
4.The triangle is a right isosceles triangle.
Observe the coordinates of the vertices.
H and I are on the same horizontal line y=4, with x-coordinates -2 and 22 respectively.
The mid-point M between H and I is therefore (10,4).
The third point J(10,-1) is therefore directly below M, and JM is therefore a median and an altitude of the triangle, with a length of 4-(-1)=5, and ∠JMH=∠JMI=90°.
This tells us that JH=JI and therefore the triangle is isosceles.
Since JM<HI, ∠JIM < 90° which implies that the triangle is obtuse.
what this suff idk
To classify the triangle, we need to analyze its side lengths and angles.
1. Obtuse isosceles triangle: This type of triangle would have one angle greater than 90 degrees and two sides that are equal in length. Let's check the side lengths of the triangle.
HI = √[(22 - (-2))^2 + (4 - 4)^2] = √(24^2) = 24
IJ = √[(10 - 22)^2 + (-1 - 4)^2] = √(12^2 + 5^2) = √(144 + 25) = √169 = 13
HJ = √[(-2 - 10)^2 + (4 - (-1))^2] = √(12^2 + 5^2) = √(144 + 25) = √169 = 13
Since not all sides are equal in length, the triangle is not an isosceles triangle.
2. Right scalene triangle: This type of triangle would have one 90-degree angle and no equal sides.
To check for a right angle, we need to determine the slopes of the sides HI, IJ, and HJ.
Slope of HI = (4 - 4) / (22 - (-2)) = 0 / 24 = 0
Slope of IJ = (-1 - 4) / (10 - 22) = (-5) / (-12) = 5/12
Slope of HJ = (4 - (-1)) / (-2 - 10) = 5 / (-12) = -5/12
None of the slopes are undefined (indicating a vertical line) or equal to zero (indicating a horizontal line), so there is no right angle in this triangle.
3. Obtuse scalene triangle: This type of triangle would have one angle greater than 90 degrees and no equal sides.
Since we already determined that not all sides are equal in length, we can now check the angles.
The easiest way to do this is by calculating the dot product of the two sides. Let's calculate the dot product of HI and IJ:
HI · IJ = (22 - (-2))(10 - 22) + (4 - 4)(-1 - 4) = (24)(-12) + (0)(-5) = -288 + 0 = -288
The dot product is negative, indicating that the angle between HI and IJ is obtuse.
Therefore, the triangle is an obtuse scalene triangle.
4. Right isosceles triangle: This type of triangle would have one 90-degree angle and two sides that are equal in length. However, we already determined that the triangle is not isosceles.
Therefore, the correct classification is option 3: The triangle is an obtuse scalene triangle.