All questions in this exercise are listed below. Click on a question to view its solution.
This exercise contains 14 questions. Use the Questions tab to view and track them.
This exercise focuses on the following concepts:
Concurrency of three or more lines (intersection at a single point)
Point of concurrency and its coordinates
Altitudes of a triangle and the orthocenter
Right bisectors (perpendicular bisectors) of triangle sides and the circumcenter
Medians of a triangle and the centroid
Area of a triangle using coordinate geometry (shoelace formula)
Collinearity condition for three points (zero area)
Determinant method for testing concurrency of lines
Below are the key formulas used in this exercise:
Area of Triangle:
Or in determinant form:
Condition for Collinearity:
Condition for Concurrency of Three Lines (, etc.):
Midpoint Formula (for medians and right bisectors):
Perpendicularity Condition (for altitudes and right bisectors):
This exercise covers coordinate geometry applications involving concurrency and area. The first part (Q1-Q8) examines concurrent lines in triangles, requiring you to find equations of altitudes (orthocenter), right bisectors (circumcenter), and medians (centroid), then verify their concurrency. The second part (Q9-Q14) focuses on calculating triangular areas using the determinant/shoelace method and applying the zero-area condition to prove collinearity or find unknown coordinates. Key strategies include: solving simultaneous equations for intersection points, using slope conditions for perpendicular lines, and recognizing that three points are collinear if and only if the triangle they form has zero area.