Given the three line segments below of lengths a b and 1 res

Given the three line segments below, of lengths a, b and 1, respectively:

construct the following length using a compass only:

Make sure to draw the appropriate diagram(s) and describe your process in words. We are also to use the following axioms and state where they are used:

1. Any two points can be connected by a line segment,

2. Any line segment can be extended to a line,

3. Any point and a line segment define a circle,

4. Points are born as intersection of lines, circles and lines and circles

Solution

Use the concept of right angles.

sqrt( (b+1)^2 - (b-1)^2 ) = 2*sqrt(b)

So with (b+1) as diameter draw a circle. With b-1 draw an arc cutting the circle. The other line segment will be twice sqrt(b). So take half the line segment by perpendicular bisector. You get sqrt(b) .

Similarly you get sqrt(a), b+sqrt(a) and the sqrt of it.

For reciprocal length we use similar triangels.

We take sqrt(b + sqrt(a)) as x

Now Let AD = x

Drw an arbitrary ray AE.

Using compass with 1 unit radius draw arc from AD to AE.

Now join D and E

Let the newly intersected point on AD be G

From G construct parallel line to DE on to AE.

We get point F

So similar triangles. AF/AE = AG/AD

AG = AE =1

So AF = 1/AD = 1/x

So constructed. All four axioms are used