Can you please explain the application of distance formula in reflection?
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Answer:
Step-by-step explanation:
The Distance Formula
Two shapes are congruent if they are exactly the same shape and exactly the same size. In congruent shapes, all corresponding sides will be the same length and all corresponding angles will be the same measure. Translations, reflections, and rotations all create congruent shapes.
If you want to determine whether two segments are the same length, you could try to use a ruler. Unfortunately, it's hard to be very precise with a ruler. You could also use geometry software, but that is not always available. If the segments are on the coordinate plane and you know their endpoints, you can use the distance formula:
d=(x2−x1)2+(y2−y1)2−−−−−−−−−−−−−−−−−−√
The distance formula helps justify congruence by proving that the sides of the preimage have the same length as the sides of the transformed image. The distance formula is derived using the Pythagorean Theorem, which you will learn more about in geometry.
Let's solve the following problems using the distance formula:
Line segment AB is translated 5 units to the right and 6 units down to produce line A′B′. The diagram below shows the endpoints of lines AB and A′B′. Prove the two line segments are congruent.
dABdABdABdABdABdAB=(x2−x1)2+(y2−y1)2−−−−−−−−−−−−−−−−−−√=(−4−3)2+(2−2)2−−−−−−−−−−−−−−−−√=(−7)2+(0)2−−−−−−−−−−√=49+0−−−−−√=49−−√=7 cmdA′B′=(x2−x1)2+(y2−y1)2−−−−−−−−−−−−−−−−−−√dA′B′=(1−8)2+(−4−(−4))2−−−−−−−−−−−−−−−−−−−√dA′B′=(−7)2+(0)2−−−−−−−−−−√dA′B′=49+0−−−−−√dA′B′=49−−√dA′B′=7 cm
Line segment AB has been rotated about the origin 90∘CCW to produce A′B′. The diagram below shows the lines AB and A′B′. Prove the two line segments are congruent.
dABdABdABdABdABdAB=(x2−x1)2+(y2−y1)2−−−−−−−−−−−−−−−−−−√=(−4−3)2+(2−2)2−−−−−−−−−−−−−−−−√=(−7)2+(0)2−−−−−−−−−−√=49+0−−−−−√=49−−√=7 cmdA′B′=(x2−x1)2+(y2−y1)2−−−−−−−−−−−−−−−−−−√dA′B′=(−2−(−2))2+(−4−3)2−−−−−−−−−−−−−−−−−−−−√dA′B′=(0)2+(−7)2−−−−−−−−−−√dA′B′=0+49−−−−−√dA′B′=49−−√dA′B′=7 cm
The square ABCD has been reflected about the line y=x to produce A′B′C′D′ as shown in the diagram below. Prove the two are congruent.
Since the figures are squares, you can conclude that all angles are the same and equal to 90∘. You can also conclude that for each square, all the sides are the same length. Therefore, all you need to verify is that mAB¯¯¯¯¯¯¯¯=mA′B′¯¯¯¯¯¯¯¯¯¯.
dABdABdABdABdABdAB=(x2−x1)2+(y2−y1)2−−−−−−−−−−−−−−−−−−√=(−6.1−(−3))2+(9.3−4.9)2−−−−−−−−−−−−−−−−−−−−−−−√=(−3.1)2+(4.4)2−−−−−−−−−−−−−√=9.61+19.36−−−−−−−−−−√=28.97−−−−√=5.38 cmdA′B′=(x2−x1)2+(y2−y1)2−−−−−−−−−−−−−−−−−−√dA′B′=(9.3−4.9)2+(−6.1−(−3))2−−−−−−−−−−−−−−−−−−−−−−−√dA′B′=(4.4)2+(−3.1)2−−−−−−−−−−−−−√dA′B′=19.36+9.61−−−−−−−−−−√dA′B′=28.97−−−−√dA′B′=5.38 cm
Since mAB¯¯¯¯¯¯¯¯=mA′B′¯¯¯¯¯¯¯¯¯¯ and both shapes are squares, all 8 sides must be the same length. Therefore, the two squares are congruent.
Examples
Given
To explain the applications of distance formula in reflection.
Distance Formula :
From the first figure, distance between A and B can be found.
If A has values and B has values
.
Distance formula is used to find the distance between any two points.
Distance ( D ) =
Reflection :
From the second figure, reflection is shown between ABCD and A'B'C'D'.
Reflection is that real image and pre-image is separated or lies above or below or lies in opposite direction of the real image.
To find distance in reflection :
From the second figure, it has four sides so the points are (A, A'), (B, B'),
(C, C'), (D, D').
Use mid-point formula to find the distance between (A, A') (B, B') and (C, C')(D, D').
Mid-point Formula :
After, using mid-point formula for (A, A') (B, B') and (C, C')(D, D'), it gives points, ( AA'BB', CC'DD').
Then, from resultant use the distance formula.
To learn more...
1. brainly.in/question/5538971
2. brainly.in/question/11987151
