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A. Dilate triangle ABC using center p and scale factor 3/2.
B. What do properties of dilations tell you about B'? (please answer B)

A Dilate triangle ABC using center p and scale factor 32 B What do properties of dilations tell you about B please answer B class=

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Part A

See the diagram below. We end up with a triangle where the sides are 3/2 = 1.5 times longer compared to the original diagram. Because segment AC is going through P (center of dilation), this means that segment A'C' will also go through P. Furthermore, it means AC and A'C' overlap. However, as mentioned earlier, A'C' is 1.5 times longer than AC.

Another thing to note: line BC is parallel to line B'C', and line AB is parallel to A'B'.

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Part B

Dilations may change the size of an object, but the shape will remain intact. This means all angles will remain the same no matter what the scale factor is.

Here's a real world example:

Cut out a triangle from a piece of paper. It can be any triangle whether right, acute, or obtuse. Then mark one of the angles (say 90 degrees). Next, hold up the triangle in front of you at arms length. Move it closer to your eye and you'll notice the triangle's size is increasing, but the marked angle of 90 degrees is not changing at all. Here the dilation is being applied with some scale factor larger than 1. When you move the triangle further away from you, the sides will shrink but the angles stay the same here as well.

So again, dilations preserve angle measures. The angles will not change.

This means angle B' is the same measure as angle B.

Ver imagen jimthompson5910

The triangle ΔA'B'C' formed following the dilation of ΔABC is a similar

triangle to ΔABC.

The correct responses are;

  • 5. a. Please find attached the drawing of the dilated triangle ΔA'B'C', created with MS Excel
  • b. The properties of dilations indicate that ∠B = ∠B'

Reasons:

5. a. With the assumption that the vertices of the triangle are;

A(0, -3), C(0, 5), and B(6, 3)

Let point P = (0, 0)

We have;

[tex]\displaystyle A' = \frac{3}{2} \times (0, -3) = \left(0, \ -\frac{9}{2} \right) = \mathbf{ (0, \, -4.5)}[/tex]

[tex]\displaystyle C' = \frac{3}{2} \times (0, \ 5) = \left(0, \ \frac{15}{2} \right) = \mathbf{(0, \, 7.5)}[/tex]

[tex]\displaystyle B' = \frac{3}{2} \times (6, \, 3) = \left(9, \ \frac{9}{2} \right) = \mathbf{ (9, \, 4.5)}[/tex]

b. From the attached diagram, and from the properties of dilation, given

that the image of ΔABC is larger than the image of ΔA'B'C' by a scale

factor of 1.5, we have that the ratio of the corresponding sides of ΔABC

and ΔA'B'C' are equal and therefore the angle formed by segment BC and

BA which is ∠B and the angle formed by segment B'C' and B'A' which is

∠B'. are equal.

[tex]\displaystyle \frac{AC}{AB} = \mathbf{\frac{A'C'}{A'B'}}[/tex]

[tex]\displaystyle \frac{AC}{sin(B)} = \frac{AB}{sin(C)}[/tex]

[tex]\displaystyle \frac{AC}{AB} = \mathbf{\frac{sin(B)}{sin(C)}}[/tex]

Similarly, we have;

[tex]\displaystyle \frac{A'C'}{A'B'} = \frac{sin(B')}{sin(C')}[/tex]

Therefore;

[tex]\displaystyle \frac{sin(B)}{sin(C)} = \mathbf{\frac{sin(B')}{sin(C')}}[/tex]

According to the properties of dilation, ∠B = ∠B'

Learn more about similar triangles here:

https://brainly.com/question/898310

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