Math Problem Statement
Solution
Let’s go through the questions one at a time:
4. If , what is the length of ?
Looking at the diagram, it appears that is a scaled version of . The scale factor can be calculated as the ratio of corresponding sides in the diagram:
In the triangle, the side labeled corresponds to the side . Based on the provided measurements:
Thus, the length of is:
5. Dilations:
(a) Dilate point using center and scale factor .
To perform the dilation of a point with respect to center and a scale factor , the new coordinates are determined by:
This means that lies on the line , and the distance from to is the distance from to . The exact position can be calculated geometrically if coordinates are known.
(b) Dilate segment using center and scale factor .
To dilate a segment with respect to a center and a scale factor , both endpoints and must be dilated individually. The new segment will have a length times the original length of .
If , the new length of after dilation is:
6. Select all the statements that must be true:
-
A. Parallelograms have four congruent sides.
This is false because only rhombuses (a type of parallelogram) have four congruent sides. -
B. Both sets of opposite sides of a parallelogram are parallel and congruent.
This is true, as it is a defining property of parallelograms. -
C. A trapezoid is a parallelogram.
This is false because trapezoids have only one pair of parallel sides, not two. -
D. Diagonals of a parallelogram bisect each other.
This is true, as it is another defining property of parallelograms. -
E. Diagonals of a parallelogram are congruent.
This is false, as only rectangles (a type of parallelogram) necessarily have congruent diagonals.
Correct Answers: B and D
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Relative Questions:
- What are some examples of shapes that involve dilation in geometry?
- How do we calculate the new coordinates of a dilated point?
- What additional properties do rhombuses and rectangles have compared to general parallelograms?
- Can a trapezoid ever have congruent diagonals?
- How does the scale factor influence both the size and position of a dilated object?
Tip:
When working with dilations, always focus on the center of dilation and the scale factor. These determine the relative size and position of the resulting shape!
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Math Problem Analysis
Mathematical Concepts
Dilations and scale factor
Proportional reasoning
Properties of parallelograms
Formulas
Dilated length = Original length × Scale factor
Dilated point = Center + Scale factor × (Point - Center)
Theorems
Properties of parallelograms
Definition of dilation
Proportionality in similar figures
Suitable Grade Level
Grades 9-10
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