What is the length of the blue line labeled r in the image below? Give your answer in terms of a and b.

blue line labeled r starting at the origin and ending at a point a,b

b|------------|
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| r / |
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_|____________|a

To find the length of the blue line labeled r, we can use the Pythagorean theorem. According to the theorem, in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

In this case, the blue line labeled r represents the hypotenuse of a right triangle, while the sides of the triangle are labeled a and b.

So, using the Pythagorean theorem, we have:

r^2 = a^2 + b^2

To find the length of r, we need to take the square root of both sides of the equation. Therefore, the length of the blue line labeled r is:

r = √(a^2 + b^2)

To find the length of the blue line labeled r, we can use the distance formula. The distance formula is derived from the Pythagorean theorem and allows us to find the distance between two points in a coordinate plane.

The formula is:

distance = √((x2 - x1)^2 + (y2 - y1)^2)

In this case, the blue line starts at the origin (0, 0) and ends at the point (a, b). Therefore, the coordinates of the two points are:

Point 1: (0, 0)
Point 2: (a, b)

We can substitute these coordinates into the distance formula:

distance = √((a - 0)^2 + (b - 0)^2)
= √(a^2 + b^2)

So, the length of the blue line labeled r is √(a^2 + b^2).

I am as frustrated as you are, because I can make very nice printer-plots as you did, but HTML here does not accept Courrier font and always suppresses multiple spaces.

To answer your question, the length of a line that goes from (0,0) to (a,b) can be calculated by the standard distance formula,
Distance = sqrt((x2-x1)^2+(y2-y1)^2)
which is based on the Pythagoras formula.
For the given case, (x1,y1)=(0,0) and (x2,y2)=(a,b) so the distance reduces to
distance=sqrt(a²+b²)