Math Problem Statement

If AB = 12, what is the length of A'B'? Dilate point C using center D and scale factor 3/4. Dilate segment AB using center D and scale factor 1/2. Select all the statements that must be true.

Solution

Let’s go through the questions one at a time:

4. If AB=12AB = 12, what is the length of ABA'B'?

Looking at the diagram, it appears that ABA'B' is a scaled version of ABAB. The scale factor can be calculated as the ratio of corresponding sides in the diagram: Scale factor=ABAB.\text{Scale factor} = \frac{A'B'}{AB}.

In the triangle, the side labeled ABA'B' corresponds to the side ABAB. Based on the provided measurements: Scale factor=612=12.\text{Scale factor} = \frac{6}{12} = \frac{1}{2}.

Thus, the length of ABA'B' is: AB=AB×Scale factor=12×12=6.A'B' = AB \times \text{Scale factor} = 12 \times \frac{1}{2} = 6.

5. Dilations:

(a) Dilate point CC using center DD and scale factor 34\frac{3}{4}.

To perform the dilation of a point CC with respect to center DD and a scale factor kk, the new coordinates CC' are determined by: C=D+k(CD).C' = D + k(C - D).

This means that CC' lies on the line DCDC, and the distance from DD to CC' is 34\frac{3}{4} the distance from DD to CC. The exact position can be calculated geometrically if coordinates are known.

(b) Dilate segment ABAB using center DD and scale factor 12\frac{1}{2}.

To dilate a segment ABAB with respect to a center DD and a scale factor k=12k = \frac{1}{2}, both endpoints AA and BB must be dilated individually. The new segment ABA'B' will have a length kk times the original length of ABAB.

If AB=12AB = 12, the new length of ABA'B' after dilation is: AB=AB×k=12×12=6.A'B' = AB \times k = 12 \times \frac{1}{2} = 6.

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:

  1. What are some examples of shapes that involve dilation in geometry?
  2. How do we calculate the new coordinates of a dilated point?
  3. What additional properties do rhombuses and rectangles have compared to general parallelograms?
  4. Can a trapezoid ever have congruent diagonals?
  5. 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