A is a scale image of figure B. Figure A x Figure B 12 2 Figure A maps to figure B with a scale factor of 7 of 아 What is the value of x?
The Correct Answer and Explanation is :
n the given problem, we have two similar figures, A and B, with a scale factor of 7. This means that each corresponding side of figure A is 7 times the length of the corresponding side of figure B.
Understanding Scale Factor and Its Implications:
he scale factor between two similar figures is the ratio of their corresponding side lengths. If the scale factor is 7, then for any corresponding sides ( a ) and ( b ) of figures A and B respectively, the relationship is:
[ \frac{a}{b} = 7 ]
his implies that each side of figure A is 7 times longer than the corresponding side of figure B.
Calculating the Ratio of Areas:
he ratio of the areas of two similar figures is the square of the scale factor. Therefore, if the scale factor is 7, the ratio of the areas ( A_{\text{A}} ) and ( A_{\text{B}} ) of figures A and B is:
[ \frac{A_{\text{A}}}{A_{\text{B}}} = 7^2 = 49 ]
his means that the area of figure A is 49 times the area of figure B.
Calculating the Ratio of Volumes:
or three-dimensional figures, the ratio of volumes is the cube of the scale factor. Thus, if the scale factor is 7, the ratio of the volumes ( V_{\text{A}} ) and ( V_{\text{B}} ) of figures A and B is:
[ \frac{V_{\text{A}}}{V_{\text{B}}} = 7^3 = 343 ]
herefore, the volume of figure A is 343 times the volume of figure B.
Application to the Given Problem:
he problem states that figure A maps to figure B with a scale factor of 7. If we are tasked with finding the value of ( x ) in the equation:
[ \text{Figure A} \times \text{Figure B} = 12 \times 2 ]
e can interpret this as the product of the areas of figures A and B. Given that the area of figure A is 49 times the area of figure B, we can set up the equation:
[ 49 \times A_{\text{B}} = 12 \times 2 ]
Solving for ( A_{\text{B}} ):
[ A_{\text{B}} = \frac{12 \times 2}{49} = \frac{24}{49} ]
herefore, the area of figure B is ( \frac{24}{49} ) square units.
Conclusion:
n summary, when two figures are similar with a scale factor of 7, the ratio of their areas is 49:1, and the ratio of their volumes is 343:1. These relationships are fundamental in geometry and are crucial for understanding the properties of similar figures.
For a visual explanation of how to find the scale factor with similar figures, you might find the following video helpful:
videoHow to Find Scale Factor with Similar Figuresturn0search6