c We're now asked about what happens to the area and perimeter if the side lengths and height are doubled. The diagram below shows the triangle with the doubled sides, along with the original triangle.
Counting the line segments with the same markers on the diagram, we can see that the perimeter doubles. To justify this observation more formally, let's use the formula for the perimeter.
P_(original)=a+b+c
P_(new)=2a+2b+2cWe can express the new perimeter in terms of the original perimeter.
When we double the sides of a triangle, the perimeter also doubles. Now, to see how the area changes, let's look at the relationship between the new and original triangles a bit differently.
We can see that four congruent copies of the original triangle make up the new triangle, so the area quadruples. To justify this observation more formally, let's use the formula for the area. Here we use that not only the sides, but the height also doubles.
A_(original)=1/2bh
A_(new)=1/2* 2b* 2h
We can express the new area in terms of the original area.
When we double the sides and height of a triangle, the area quadruples.
d Now we are supposing that the sides and the height were divided by 3. Let's use the formula for the perimeter, and modify it to represent the division.
P_(original)=a+b+c
P_(new)=1/3a+1/3b+1/3c
We can express the new perimeter in terms of the original perimeter.
When the side lengths of the triangle are divided by three, the perimeter is also divided by three. Now, to see what happens to the area, let's use the formula for the area. Here we use that not only the side lengths, but the height is also divided by three.
A_(original)=1/2bh
A_(new)=1/2* 1/3b* 1/3h
We can express the new area in terms of the original area.
