Math Problem Statement
Solution
The problem in the image is asking to find the value of for which , meaning that the triangle has a right angle at point .
The points of the triangle are:
To solve this, we need to use the concept that two lines are perpendicular if the product of their slopes is equal to .
Steps:
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Find the slope of line : The slope of a line between two points and is given by: For and , the slope of is:
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Find the slope of line : Using the points and , the slope of is:
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Set up the perpendicularity condition: Since , the lines and must be perpendicular, meaning: Substituting the slopes:
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Solve for : Now solve the equation: Multiply both sides by : Simplify the right-hand side: Distribute the negative sign: Bring all terms to one side: Now, solve this quadratic equation using the quadratic formula: So, the two possible values for are:
Would you like more details on the steps or another part of the solution explained?
Here are some related questions for further practice:
- What is the equation of a line passing through points and ?
- How can we verify that the triangle is right-angled using the distance formula?
- What is the area of the triangle formed by points , , and ?
- How does the value of affect the orientation of the triangle?
- If , how would we find the value of ?
Tip: Always check the perpendicularity condition by verifying the product of the slopes when solving geometry problems involving right angles.
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Math Problem Analysis
Mathematical Concepts
Geometry
Coordinate Geometry
Perpendicular Slopes
Formulas
Slope formula: m = (y2 - y1) / (x2 - x1)
Perpendicular lines condition: m1 * m2 = -1
Quadratic equation: ax^2 + bx + c = 0
Theorems
Perpendicular slope theorem
Quadratic formula
Suitable Grade Level
Grades 9-10
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