Is a triangle with vertices G(3,2), H(8,5) and K(0,10) a right angle. explain plss, pls. very imp.
I would make a sketch on a graph to see what the problem is
find the length of GH, GK and HK
then test using Pythagoras
I will find GH, you find the other two.
GH = √(8-3)^2 + (5-2)^2 )
= √(25 + 9) = √34
hint: remember that if you square √34 , you get 34
To determine if the triangle with vertices G(3,2), H(8,5), and K(0,10) is a right triangle, we need to calculate the slopes of its sides and check if any two sides have a negative reciprocal slope.
The slope between two points (x1, y1) and (x2, y2) can be calculated using the formula:
slope = (y2 - y1) / (x2 - x1)
Let's calculate the slopes of the sides GH, GK, and HK:
Slope of GH:
GH = (5 - 2) / (8 - 3) = 3 / 5
Slope of GK:
GK = (10 - 2) / (0 - 3) = 8 / -3 = -8/3
Slope of HK:
HK = (10 - 5) / (0 - 8) = 5 / -8 = -5/8
Now, we need to check if any two sides have negative reciprocal slopes. If they do, it means they are perpendicular to each other, and thus, the triangle is a right triangle.
Let's check:
- The slopes GH and GK are not negative reciprocals of each other.
GH = 3/5 and GK = -8/3 are not negative reciprocals.
- The slopes GH and HK are not negative reciprocals of each other.
GH = 3/5 and HK = -5/8 are not negative reciprocals.
- The slopes GK and HK are not negative reciprocals of each other.
GK = -8/3 and HK = -5/8 are not negative reciprocals.
Since none of the slopes of the sides GH, GK, and HK are negative reciprocals, it means the triangle with vertices G(3,2), H(8,5), and K(0,10) is not a right triangle.
Therefore, the triangle is not a right angle.
To determine whether a triangle is a right angle triangle, we need to check if any of the three angles in the triangle is a right angle, which is 90 degrees.
To do this, we can use the Pythagorean theorem, which states that in a right angle triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
Let's consider the lengths of the sides of triangle GHK:
- Side GH: The length of GH can be found using the distance formula, which is the square root of the sum of the squares of the differences between the x-coordinates and y-coordinates of the vertices.
Length of GH = sqrt((8-3)^2 + (5-2)^2) = sqrt(25 + 9) = sqrt(34)
- Side GK: The length of GK can be found similarly:
Length of GK = sqrt((0-3)^2 + (10-2)^2) = sqrt(9 + 64) = sqrt(73)
- Side HK: The length of HK can be found in the same way:
Length of HK = sqrt((8-0)^2 + (5-10)^2) = sqrt(64 + 25) = sqrt(89)
Now, let's compare the squares of the lengths of the sides.
- (Length of GH)^2 = (sqrt(34))^2 = 34
- (Length of GK)^2 = (sqrt(73))^2 = 73
- (Length of HK)^2 = (sqrt(89))^2 = 89
Since none of the squares of the lengths of the sides equals the sum of the squares of the other two sides, the triangle GHK is not a right angle triangle.
Therefore, the triangle with vertices G(3,2), H(8,5), and K(0,10) is not a right angle triangle.