How many positive factors does 2023 have?

Answer: choice D) 18

Explanation:

M is the midpoint of QR, so QM = MR, meaning that the two smaller pieces are equal (the two pieces combine back to QR)

Use substitution and isolate x

QM = MR

2x+5 = 5x-1

2x-5x = -1-5

-3x = -6

x = -6/(-3)

x = 2

If x = 2, then QM is

QM = 2x+5 = 2*2+5 = 9

and MR is

MR = 5x-1 = 5*2-1 = 9

both are 9 units long. Both are the same length as expected, so this helps confirm the answer.

The last thing we do is add up the two pieces QM and MR to get the length of QR. This is using the segment addition postulate.

QR = QM + MR

QR = 9 + 9

QR = 18

note: a negative length is never possible in any situation, so you can immediately eliminate choice B from the list of possible answers without even doing any math.

Answer:

I. A' =  (5,-5), B' = (-1,6), C' = (6,1)

II. A' =  (5,-11), B' = (-1,0), C' = (6,-5)

III. A' = (-5,-2), B = (6,4), C' = (1,-3)

Step-by-step explanation:

We are given the vertices of ΔABC as A = (2,-5), B = (-4,6) and C = (3,1).

I. It is required to 'reflect the triangle about the line x= 3'.

This rule changes (x,y) to (x+3,y).

So, the new vertices are given by,

A' = (2+3,-5) = (5,-5)

B' = (-4+3,6) = (-1,6)

C' = (3+3,1) = (6,1)

II. It is required to 'translate the triangle 3 units to the right and 6 units down'.

This rule changes (x,y) to (x+3,y-6).

So, the new vertices are given by,

A' = (2+3,-5-6) = (5,-11)

B' = (-4+3,6-6) = (-1,0)

C' = (3+3,1-6) = (6,-5)

III. It is required to 'rotate the triangle by 90° about the origin counter-clockwise'.

This rule changes (x,y) to (y,-x).

So, the new vertices are given by,

A = (2,-5) implies A' = (-5,-2)

B = (-4,6) implies B = (6,4)

C = (3,1) implies C' = (1,-3)