Coordinate Geometry

Concept: In a 2D Cartesian coordinate system, a point P is represented by an ordered pair .
The -coordinate represents the perpendicular distance of the point from the y-axis.
The -coordinate represents the perpendicular distance of the point from the x-axis. Distance is always a non-negative value.

Step 1: Identify the coordinates of the point P The given point is P (6,8). Here, and .

Step 2: Understand "perpendicular distance from the x-axis" The x-axis is the horizontal line where . The perpendicular distance of a point from the x-axis is the length of the vertical line segment from the point down to (or up to) the x-axis. This length is simply the absolute value of the y-coordinate. Imagine plotting the point P(6,8):
You move 6 units to the right along the x-axis.
Then, you move 8 units up, parallel to the y-axis. The distance you moved upwards (8 units) is the perpendicular distance from the x-axis.

Step 3: Determine the perpendicular distance For the point P(6,8), the y-coordinate is 8. The perpendicular distance from the x-axis is units.

Step 4: Compare with the options The calculated distance is 8. This matches option (1).

Step 1: Identify the type of triangle.
The given vertices are , , and .
The vertex is the origin.
The vertex lies on the x-axis.
The vertex lies on the y-axis.
Since two sides of the triangle lie along the x and y axes, this is a right-angled triangle.
Step 2: Determine the base and height of the triangle.
The length of the base can be calculated as the distance between and along the x-axis.
Base ( ) = units.
The height of the triangle can be calculated as the distance between and along the y-axis.
Height ( ) = units.
Step 3: Calculate the area of the triangle.
The formula for the area of a triangle is .
Area =
Area =
Area =
Area = Square units.
Alternatively, using the determinant formula for the area of a triangle with vertices , , and :
Area = Let , , and .
Area =
Area =
Area =
Area = Square units.
Step 4: Compare with the given options.
The calculated area is Square units, which matches option (4). (4) 15 Square units