Length of a line segment joining two points, coordinates of a point P that lies on a line segment and divides the line segment in a given ratio, computing the slope of a line if the coordinates of two points through which the line passes is known, testing for collinearity of three or more points, finding the equation of a line and that of parallel and perpendicular lines, computing coordinates of centroid, orthocenter and circumcenter of a triangle, finding equations of median, perpendicular bisector, altitude and angle bisector of a triangle coordinates of whose vertices are known, finding radius of a circumscribing circle, equations of circles, and distance between a point and a line.
A straight line 4x + 3y = 24 forms a triangle with the coordinate axes. What is the distance between the orthocentre of the triangle and the centre of the circle that circumscribes the triangle?
Step 1: Compute the coordinates of the Triangle.
Step 2: Determine the coordinates of the orthocentre.
Step 3: Determine the coordinates of the circumcentre.
Step 4: Compute the length of the line segment.
Quantitative Comparison
Quantity A | Quantity B |
---|---|
Length of the segment of the line 4x + 3y = 12 intercepted between the coordinate axes. | Length of the median to side BC of triangle whose coordinates are A(4, 4), B(10, 4) and C(4, 12) |
Step 1: Quantity A: Compute the length of the line segment joining the x and y intercepts of 4x + 3y = 12.
Step 2: Quantity B: Compute the coordinates of the mid point D.
Step 3: Quantity B: Compute the length of the median AD.
Step 4: The Comparison: Compare the length of the line segment joining x and y intercepts of 4x + 3y = 12 and the median AD of the triangle given in Quantity B.
Points C and D trisect the line segment joining points A(4, 5) and B (16, 14). What is the length of the line segment CD?
Step 1: Compute the length of the line segment AB using the distance formula.
Step 2: Because C and D trisect AB, AC = CD = DB.
Step 3: Hence, 1/3rd length of AB = CD.
What is the slope of a line that makes an intercept of 5 in the positive direction of y axis and 1 in the negative direction of x axis?
Step 1: Slope of a line = -(y-intercept)/(x-intercept).
Step 2: Substitute values given in the question and compute slope.
Which of the following could be the slope of the line that passes through the point (4, 5) and intercepts the y-axis below the origin?
Indicate all that apply.
Step 1: Slope of a line given two points (x1, y1) and (x2, y2) through which the line passes = (y2 - y1)/(x2 - x1).
Step 2: The line passes through (4, 5) and (0, y1) where y1 is the y-intercept of the line.
Step 3: It is given that y1 is negative. Therefore, (5 - y1) will be greater than 5.
Step 4: Compute the range of values that the slope can take and identify answer options that satisfy the condition.
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