Question

Triangle NOP, with vertices N(3,3), O(7,5), and P(4,9), is drawn inside a rectangle.What is the area, in square units, of triangle NOP?

Answer 1

To find the area of triangle NOP, we can use the formula for the area of a triangle:

Area = 1/2 * base * height

First, we need to calculate the base of the triangle NOP. The base can be calculated as the distance between points N(3,3) and O(7,5). Using the distance formula:

Base = √[(7-3)^2 + (5-3)^2]
Base = √[4^2 + 2^2]
Base = √(16 + 4)
Base = √20
Base ≈ 4.47 units

Next, we need to calculate the height of triangle NOP. The height can be calculated as the perpendicular distance from point P(4,9) to the line containing the base NO. Since the line containing NO is parallel to the y-axis, the height is the difference in y-coordinates between points P and any point on the line containing NO. So, the height is the difference between the y-coordinate of point P(4,9) and the y-coordinate of point N(3,3):

Height = 9 - 3
Height = 6 units

Now, we can calculate the area of triangle NOP:

Area = 1/2 * base * height
Area = 1/2 * 4.47 * 6
Area ≈ 13.41 square units

Therefore, the area of triangle NOP is approximately 13.41 square units.