MHB What is the perimeter of triangle ABC?

[mathdad]

The vertices of triangle ABC are A(1, 1), B(9, 3), and
C(3, 5).

1. Find the perimeter of triangle ABC.

I must use the distance formula for points on the xy-plane to find all three sides. I then add all three sides. Correct?

2. Find the perimeter of the triangle that is formed by joining the midpoints of the three sides of triangle ABC.

I am not too sure about part 2.

3. Compute the ratio of the perimeter in part 1 to the perimeter in part 2.

I will let R = ratio.

The set up for part 3 is

R = (perimeter of part 1)/(perimeter of part 2)

Correct? I cannot do part 3 without computing part 2, which I don't know how to do.

 

[Greg]

The vertices of triangle ABC are A(1, 1), B(9, 3), and
C(3, 5).

1. Find the perimeter of triangle ABC.

I must use the distance formula for points on the xy-plane to find all three sides. I then add all three sides. Correct?

Correct.

2. Find the perimeter of the triangle that is formed by joining the midpoints of the three sides of triangle ABC.

A sheet of graph paper will be handy for this exercise. Mathematically, the coordinates of a midpoint may be found with

$$x_M=\frac{x_1+x_2}{2},\quad y_M=\frac{y_1+y_2}{2}$$

3. Compute the ratio of the perimeter in part 1 to the perimeter in part 2.

I will let R = ratio.

The set up for part 3 is

R = (perimeter of part 1)/(perimeter of part 2)

Correct?

Correct. You may also write

$$P_1:P_2$$

As a hint, the desired ratio is 2:1.

 

[mathdad]

1. Use the distance formula to find the distance between all 3 sides. Add all three sides. Adding all three sides yields perimeter 1.

2. Use the midpoint formula to find the midpoint of the distance between the three given points.

3. Find the distance between the 3 midpoints found in part 2 above. Add all 3 sides. This yields perimeter 2.

4. The ratio = (perimeter 1)/(perimeter 2)

This exercise is related more to geometry mixed with algebra. Correct? I did not post in the geometry forum because the question is from David Cohen's precalculus textbook.