What is the area of a triangle whose vertices are R(−4, 2) , S(1, 2) , and T(−5, −4)
To find the area of a triangle given its vertices, we can use the formula for the area of a triangle:
Area = ½ * base * height
In order to use this formula, we need to determine the base and height of the triangle.
Step 1: Find the length of one of the sides of the triangle.
Let's consider the side RS. We can use the distance formula to find its length.
Distance formula: d = √((x₂ - x₁)² + (y₂ - y₁)²)
Using the coordinates of R(−4, 2) and S(1, 2):
d₁ = √((1 - (-4))² + (2 - 2)²)
= √(5² + 0²)
= √25
= 5
So, the length of side RS is 5.
Step 2: Find the height of the triangle.
To find the height, we need to draw an altitude from one of the vertices to the opposite side. In this case, let's draw an altitude from vertex T to side RS.
The length of the altitude is equal to the perpendicular distance between vertex T and side RS. We can calculate this distance using the formula for the distance between a point and a line.
Distance from point T to line RS:
d₂ = |(x₂ - x₁) * (y₁ - y₃) - (x₁ - x₃) * (y₂ - y₁)| / √((x₂ - x₁)² + (y₂ - y₁)²)
Using the coordinates of T(−5, −4):
d₂ = |((1 - (-4)) * (2 - (-4)) - (-4 - (-5)) * (2 - 2))| / √((1 - (-4))² + (2 - 2)²)
= |-3 * 6 - 1 * 0| / √(5² + 0²)
= |-18| / 5
= 18 / 5
= 3.6
So, the height of the triangle is 3.6.
Step 3: Calculate the area of the triangle.
Now that we have the base (RS = 5) and the height (3.6), we can use the area formula to compute the area.
Area = ½ * base * height
= ½ * 5 * 3.6
= 9 square units
Therefore, the area of the triangle is 9 square units.
It always helps to make a sketch before you start a problem.
In this case, notice that RS is a horizontal line, so it is easy to that RS = 5
let's make RS the base, then since T is 6 units below that line, the area
= (1/2)(base)(height
= ....