In the figure shown two lines intersect at a right angle and two semicircles are drawn so that each semicircle has its diameter on one line and is tangent to the other line. The larger semicircle has radius of 1. The smaller semicircle intersects the larger one, dividing the larger semicircle arc into the ratio 1:5. What is the radius of the smaller semicircle? Express your answer in simplest radical form.

  2-√3

Step-by-step explanation:

In the attached, we have labeled some points so they are easier to talk about.

We are given that arc AG : arc GC = 1 : 5, so the measure of arc AG is 1/6 of the total measure of arc AC. That is, AG = 180°/6 = 30°.

Inscribed angle ACG is half that measure, so is 15°. Triangle BEA is similar to and half the size of triangle CDA, so angle ABE is also 15°. Radius AB is 1, so side AE is tan(15°).

We can use the half-angle formula for tangent to find the measure of AE.

 

The radius of the smaller circle is 2-√3.

Answer: question no. 1 -a. yes, the sides are in the ratio 2 : 5.question no. 2 -b. 59°question no. 3 -c. angles x and y are remote interior angles.question no. 4 -d. 90°, 45° and 45°question no. 5 -b. mn and qrstep-by-step explanation: question no. 1 -rs : xu = 3 : 7.5 = 2 : 5rt : x? = 6.4 : 16 = 2 : 5st : u? = 6 : 15 = 2 : 5as the ratios of all sides are equivalent, the triangles must be congruent and the sides must be proportional.therefore, the answer is: a. yes, the sides are in the ratio 2 : 5.question no. 2 -in the second diagram, angle b is the only letter not shown. therefore, the number must be equivalent to angle b.if we substitute the value of angle b as being 22° in the first diagram, we can now use the angle sum of a triangle to obtain the value of angle a.angle sum of a triangle = 180°therefore: let angle a = aa + 99 + 22 = 180a = 180 - 99 - 22a = 59°therefore, the answer is: b. 59°question no. 3 -angles x and y don't share the vertex of the triangle with the exterior angle w. remote interior angles don't share a vertex of a triangle with an exterior angle. this means that angles x and y are remote interior angles.therefore, the answer is: c. angles x and y are remote interior angles.question no. 4 -the sum of all interior angles of a triangle must be equivalent to 180°.90°, 45° and 45° are the only set of angle measures that are equivalent to 180°.therefore, the answer is: d. 90°, 45° and 45°question no. 5 -the only pair of sides stated in the options which are corresponding are mn and qr.therefore, the answer is: b. mn and qr.