Exercise 4 Page 207

a We will use the same triangle

△ A B C

as was used in the Exploration in the book. Use a dynamic geometry software, for example Geogebra, to graph it.

Instead of the center $D$ we will use the vertex $A$ as the center. With the scale factor $2,$ the dilated triangle will have the same vertex $A$ as the original triangle but twice as long sides. We will use our software to draw the new triangle.

We should now compare the coordinates, segments, and angles for the figures. The coordinate $A$ and $A^{'}$ are the same since its the center of dilation. We will compare the coordinates of $B$ and $C$ with $B^{'}$ and $C^{'} .$

Vertex	Dilated	Original	Difference
$B$	$(0, 5)$	$(1, 3)$	$(- 1, 2)$
$C$	$(4, 3)$	$(3, 2)$	$(1, 1)$

The vertex $B$ moved $1$ unit on the $x -$ axis and $2$ units on the $y -$ axis. The vertex $C,$ on the on the other hand, moved $1$ unit on each axis after the dilation. Looking at the segments lengths they increased by a factor $2,$ the same as the scale factor. The angles are the same for both triangles.

b We will follow the same steps as we did in Part A. We will enter the coordinates into our software for

△ A B C

and then we will scale it with the factor

\frac{1}{2} .

Now we will start by comparing the coordinates. Since the center is $A$ the coordinates will be the same for $A$ and $A^{'} .$ Instead, we compare the difference between the $△ A B C$ and $△ A^{'} B^{'} C$ for the other vertices.

Vertex	Dilated	Original	Difference
$B$	$(1.5, 2)$	$(1, 3)$	$(0.5, - 1)$
$C$	$(2.5, 1.5)$	$(3, 2)$	$(- 0.5, - 0.5)$

The coordinates are moved differently for each vertex. When comparing the length, we see that they are half the size of the original triangle. The angles are the same after the dilation.

c If we look at the coordinates for the dilated vertices of Part A and Part B, it is hard to make a conclusion. The coordinates of

A

did not change while the others did change.

Vertex	Difference $(a)$	Difference $(b)$
$B$	$(- 1, 2)$	$(0.5, - 1)$
$C$	$(1, 1)$	$(- 0.5, - 0.5)$

The side lengths changed by the given factors, becoming twice as long in Part A and half as long in Part B. The angle measures remained the same in both cases.

Subchapter links

Communicate Your Answer p.207

Monitoring Progress p.208-211

Exercises p.212-214