Math Problem Statement
解图中的数学题
Solution
The problem in the image involves finding the area of the shaded region in a unit square where and is the midpoint of .
Let's break down the problem step-by-step.
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Identify and label the points:
- is at
- is at
- is at
- is at
- is the midpoint of , so is at
- , which means divides in the ratio 2:1. Given that is at and is at , is at
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Find the coordinates of , the intersection point of and :
To find , we need the equations of lines and :
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Equation of : The line passes through and . The slope is . Using the point-slope form, the equation is .
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Equation of : The line passes through and . The slope is . Using the point-slope form, the equation is .
Setting the equations equal to each other to find the intersection: Solving for :
3 = 2.5x \\ x = \frac{6}{5}$$ Substituting $$x = \frac{6}{5}$$ back into $$y = -0.5x + 1$$: $$y = -0.5 \times \frac{6}{5} + 1 = 1 - 0.6 = 0.4$$ Therefore, the coordinates of $$G$$ are $$\left(\frac{6}{5}, 0.4\right)$$. -
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Calculate the area of the shaded region :
Using the formula for the area of a triangle given its vertices , , :
Substituting the coordinates of , , and :
The area of the shaded region is square units.
Would you like any further details or have any questions?
Here are five more questions for you to consider:
- How do you calculate the intersection of two lines algebraically?
- Can you find the area of a triangle given its vertices?
- What are the different forms of a line equation?
- How do you determine the coordinates of a point that divides a line segment in a given ratio?
- What is the significance of the slope of a line?
Tip: When solving geometry problems, drawing a clear diagram and labeling all known points and lines can greatly simplify the solution process.
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Math Problem Analysis
Mathematical Concepts
Geometry
Coordinate Geometry
Area Calculation
Formulas
Area of a triangle given vertices
Theorems
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Suitable Grade Level
High School
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