i have a question in math
(( distance between two points ))
the vertices of a right triangle are S(-2,-2), T(10,-2), and R(4,5). find the area of the triangle.
please in detales .. thank u
first of all, your triangle is not right-angled.
The slopes are 7/6, -7/6 and zero
None of these are negative reciprocals of each other, a condition for a right angle.
But the area is easy to find
Clearly the base ST = 12 units and the height from R to ST is 7
So the area = 1/2 x 12 x 7
= 42 units^2
thank u
can u explain how u get the height ?
To find the area of a triangle, we can use the formula called "Heron's formula" or we can use the formula for the area of a triangle given the lengths of two sides and the included angle. In this case, we have the coordinates of the vertices of the triangle, so we need to use the formula for the area of a triangle given the coordinates of its vertices.
The first step is to find the lengths of the sides of the triangle. We can use the distance formula to determine the lengths. The distance formula is calculated as:
d = √((x2 - x1)^2 + (y2 - y1)^2)
Let's calculate the lengths of the sides ST, SR, and RT:
ST = √((10 - (-2))^2 + (-2 - (-2))^2)
= √((12)^2 + (0)^2)
= √(144 + 0)
= √144
= 12
SR = √((4 - (-2))^2 + (5 - (-2))^2)
= √((6)^2 + (7)^2)
= √(36 + 49)
= √85
RT = √((10 - 4)^2 + (-2 - 5)^2)
= √((6)^2 + (-7)^2)
= √(36 + 49)
= √85
Now that we have the lengths of the sides, we can use the formula for the area of a triangle given the lengths of two sides and the included angle. In this case, we don't have the included angle, but we can use the coordinates to find it using trigonometry. The included angle can be found using the inverse tangent function:
θ = arctan((y2 - y1)/(x2 - x1))
In this case, we can find the included angle θ between ST and SR using the coordinates of S and T:
θ = arctan((-2 - (-2))/(10 - (-2)))
= arctan(0/12)
= arctan(0)
= 0
Since the included angle is 0 degrees, we can use the formula for the area of a triangle given the lengths of two sides:
Area = (1/2) * ST * SR * sin(θ)
In this case, sin(0) = 0, so the formula simplifies to:
Area = (1/2) * ST * SR * 0
= 0
Therefore, the area of the right triangle with vertices S(-2,-2), T(10,-2), and R(4,5) is 0.