The vertices of a triangle are A(1,2), B(4,2), and C (4,6). What is the length of the hypotenuse?
Very confused HELP!!!
To find the length of the hypotenuse of a triangle, we need to determine which side of the triangle is the hypotenuse. The hypotenuse is always the side opposite the right angle in a right-angled triangle.
In this case, we need to check if the triangle formed by the given vertices is a right-angled triangle. We can do this by calculating the slopes of the lines formed by connecting the three pairs of vertices.
The slope of a line can be calculated using the formula:
m = (y2 - y1) / (x2 - x1)
Let's calculate the slope of side AB and AC:
Slope of AB:
m_AB = (y2 - y1) / (x2 - x1)
= (2 - 2) / (4 - 1)
= 0 / 3
= 0
Slope of AC:
m_AC = (y2 - y1) / (x2 - x1)
= (6 - 2) / (4 - 1)
= 4 / 3
Now, if the product of the slopes of two sides of a triangle is -1, then the triangle is a right-angled triangle. Let's check if the slope of AB and AC satisfies this condition:
m_AB × m_AC = 0 × (4 / 3)
= 0
Since the product is zero (not -1), we can conclude that the triangle ABC is not a right-angled triangle.
Therefore, we cannot use the Pythagorean theorem to directly calculate the hypotenuse. Instead, we can calculate the lengths of the three sides of the triangle using the distance formula:
d = √((x2 - x1)^2 + (y2 - y1)^2)
Now, let's calculate the lengths of the sides AB, BC, and AC:
Length of AB:
d_AB = √((4 - 1)^2 + (2 - 2)^2)
= √(3^2 + 0^2)
= √(9 + 0)
= √9
= 3
Length of BC:
d_BC = √((4 - 4)^2 + (6 - 2)^2)
= √(0^2 + 4^2)
= √(0 + 16)
= √16
= 4
Length of AC:
d_AC = √((4 - 1)^2 + (6 - 2)^2)
= √(3^2 + 4^2)
= √(9 + 16)
= √25
= 5
Therefore, in this case, the triangle ABC is not a right-angled triangle, and we cannot determine the length of the hypotenuse since there is no hypotenuse.
hyp. = AC = sqrt((4-1)^2+(6-2)^2) =
A and B have the same y-coordinate ... AB parallel to x-axis
B and C have the same x-coordinate ... BC parallel to y-axis
AB and BC are perpendicular ... AC is the hypotenuse
... looks like a 3-4-5 triangle