Question

6. In the diagram below, suppose that point M and point N are the midpoints of PQ and RQ, respectively, and PN = RM.a. Find the coordinates of point M and Point Nb. Using the distance formula) write and simplify an equation expressing the fact that PN=RM.c. Use the equation you found in part (b) to express b in terms of a alone (solve for b) What does this tell you about Triangle PQR?

Answer 1

a)

The formula for calculating the coordinates midpoint between two points is expressed as

midpoint = (x1 + x2)/2, (y1 + y2)/2

Considering point M, the coordinates of the endpoints are

P(0, 0), Q(2b, 2c)

Considering point N, the coordinates of the endpoints are

R(2a, 0), Q(2b, 2c)

Finding the midpoint of PQ or coordinates of point M, we have

x1 = 0, y1 = 0

x2 = 2b, y2 = 2c

M = (0 + 2b)/2, (0 + 2c)/2 = 2b/2, 2c/2

M = (b,c)

Finding the midpoint of RQ or coordinates of point N, we have

x1 = 2a, y1 = 0

x2 = 2b, y2 = 2c

N = (2a + 2b)/2, (0 + 2c)/2 = 2(a + b)/2, 2c/2

N = ((a + b), c))

b)

The formula for calculating the distance beween two points is expressed as

`\text{distance = }\sqrt[]{(x2-x1)^2+(y2-y1)^2}`

For PN, the coordinates are P(0, 0), N(a + b), c

x1 = 0, y1 = 0

x2 = a + b, y2 = c

Thus,

`\begin{gathered} PN\text{ = }\sqrt[]{(a+b-0)^2+(c-0)^2}\text{ = }\sqrt[]{(a+b)^2+c^2} \\ PN\text{ = }\sqrt[]{a^2+2ab+b^2+c^2} \\ PN\text{ = }\sqrt[]{b^2+2ab+a^2+c^2} \end{gathered}`

For RM, the coordinates are R(2a, 0), M(b, c)

x1 = 2a, y1 = 0

x2 = b, y2 = c

`RM\text{ = }\sqrt[]{(b-2a)^2+(c-0)^2}=\sqrt[]{(b-2a)^2-x^2}`