The figure below shows ∆XYZ graphed on the coordinate plane. What is the area of ∆XYZ
X 0,8
Y 6,8
Z 5,3
It's a triangle but it has a irregular shape.
To find the area of the triangle ∆XYZ, we can use the formula for the area of a triangle using the coordinates of the three points.
We can first calculate the base of the triangle by finding the distance between points X and Y, which is 6 units.
Next, we need to find the height of the triangle. To do this, we can draw a perpendicular line from point Z to the line XY and find the length of this line. We can see that this forms a right triangle with legs of length 3 and 5. Using the Pythagorean theorem, we can calculate the length of the hypotenuse (the height) as follows:
h = √(3^2 + 5^2)
h = √(9 + 25)
h = √34
Now that we have the base and height of the triangle, we can calculate the area using the formula:
Area = 1/2 * base * height
Area = 1/2 * 6 * √34
Area = 3 * √34
Area ≈ 18.47 square units
Therefore, the area of ∆XYZ in the given coordinates is approximately 18.47 square units.
The possible answers are
15 sq. units
15√2 sq. units
30 sq. units
30√2 sq. units
To determine the area of ∆XYZ, we will use the formula for the area of a triangle given its coordinates. However, before we proceed with finding the area, let's first determine the coordinates of the three vertices of the triangle based on the information provided.
Given the coordinates of the three points:
X (0, 8)
Y (6, 8)
Z (5, 3)
We can see that the base of the triangle ∆XYZ is XY. Using the coordinates of X and Y, we find that the base XY has a length of 6 units.
Next, we need to find the height of the triangle, which can be accomplished by calculating the vertical distance between point Z and the line XY. The perpendicular distance from Z to XY will give us the height of the triangle.
By drawing the perpendicular from Z to XY, we can determine that the length of the height is 5 units.
Now, we can use the formula for the area of a triangle:
Area = 1/2 * base * height
Area = 1/2 * 6 * 5
Area = 15 square units
The area of ∆XYZ with the given coordinates is 15 square units.
Therefore, the correct answer is: 15 sq. units.