Math Problem Statement
Solution
Let's go through each part of the problem systematically.
Given:
Vertices of the triangle PQR:
(a) Find the equation of the median RM
A median of a triangle is a line segment that joins a vertex to the midpoint of the opposite side.
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Midpoint M of PQ: The midpoint formula is: For and :
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Equation of RM: Now, we know that point and point lie on the median. We can find the equation of the line using the point-slope form: where is the slope of the line, and is a point on the line. The slope between and is: Using point : Expanding: So, the equation of the median is:
(b) Find the equation of the altitude AP
An altitude of a triangle is a line segment from a vertex perpendicular to the opposite side.
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Slope of side QR: The slope of can be calculated as:
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Slope of the perpendicular line (altitude AP): The slope of a line perpendicular to another line is the negative reciprocal of the slope of the other line. Since the slope of is 1, the slope of the perpendicular line is:
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Equation of AP: Using the point-slope form again with point and slope : Expanding: So, the equation of the altitude is:
(c) Find the coordinates of the point of intersection of RM and AP
To find the point of intersection of the two lines, solve the system of equations:
- (from the median RM)
- (from the altitude AP)
Set the equations equal to each other: Solving for :
Now, substitute into either equation to find . Using :
Thus, the point of intersection is:
Summary of Answers:
(a) Equation of the median :
(b) Equation of the altitude :
(c) Point of intersection of RM and AP:
Would you like any further clarification or more details? Here are five related questions:
- How can you find the centroid of a triangle?
- What is the formula to calculate the area of a triangle given its vertices?
- Can you determine the length of the median in this triangle?
- How do you find the orthocenter of a triangle?
- What properties does the intersection of medians have in a triangle?
Tip: In geometry, intersections of special lines like medians and altitudes often give key points (centroid, orthocenter), useful in various applications of triangle properties!
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Math Problem Analysis
Mathematical Concepts
Coordinate Geometry
Equations of Lines
Triangles
Medians
Altitudes
Formulas
Midpoint formula: M = ((x1 + x2)/2, (y1 + y2)/2)
Slope formula: m = (y2 - y1) / (x2 - x1)
Point-slope form of a line: y - y1 = m(x - x1)
Theorems
Basic Theorems of Coordinate Geometry
Properties of Medians and Altitudes in a Triangle
Suitable Grade Level
Grades 9-12
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