We are asked to use a to draw two different that are to the given one. Then, we want to find if the of the corresponding heights are to the ratios of the corresponding side lengths. Let's do these things, one at a time.
Drawing the Triangles
Let's take a look at the given .
Recall that a is a or a sequence of dilations and . Let's dilate the given triangle in two different ways to get two different isosceles triangles that are similar to the given one. To dilate a figure with respect to the , we multiply the of each by the k.
(x,y) → (kx,ky)
Let's identify the coordinates of the vertices of the given figure. We will also measure the height and sides.
For the first triangle, let's choose k= 2.
| (x,y)
|
(2x,2y)
|
Simplify
|
| (0,0)
|
( 2*0, 2*0)
|
(0,0)
|
| (3,4)
|
( 2*3, 2*4)
|
(6,8)
|
| (6,0)
|
( 2*6, 2*0)
|
(12,0)
|
We can plot the new vertices on the and connect them using a ruler to draw the first triangle. We will also measure its height and sides.
For the second triangle, let's choose k= 0.5.
| (x,y)
|
(0.5x,0.5y)
|
Simplify
|
| (0,0)
|
( 0.5*0, 0.5*0)
|
(0,0)
|
| (3,4)
|
( 0.5*3, 0.5*4)
|
(1.5,2)
|
| (6,0)
|
( 0.5*6, 0.5*0)
|
(3,0)
|
Again, we can plot the new vertices on the coordinate plane, connect them, and measure its height and sides.
Checking the Ratios
Now we will consider the ratios of the original triangle and the new triangles. Let's find the ratios of the corresponding side lengths of the original triangle and the first similar triangle that we drew. The lengths of the sides in the original triangle are 5 and 6. The corresponding side lengths in the first similar triangle are 10 and 12.
5/10 = 6/12 = 1/2
The height of the original triangle is 4 and the height of the first similar triangle is 8.
4/8 = 1/2
We can see that the ratios are equivalent. Now, let's check the ratios for the second similar triangle that we drew. The corresponding side lengths to the sides of measures 5 and 6 are 2.5 and 3.
5/2.5 = 6/3 = 2
The height of the original triangle is 4 and the height of the second similar triangle is 2.
4/2 = 2
We can see that the ratios of the corresponding heights are equivalent to the ratios of the corresponding side lengths.
