Three concentric circles have radii 5, 10, 15. Chords of the 2 larger circles are tangent to the 2 smaller circles, respectively, and the points of tangency lie on the same radius of the largest circle. Find the area of the trapezoidal region determined by the endpoints of the chords.

Area of the trapezoidal region = 99 square units

Step-by-step explanation:

Let the common center of the three circles be O and OA to be the radius of the larger circle, intersecting the smallest circle and middle circle at B and C, respectively.

Where DE be the chord of the middle circle and FG be the chord of the largest circle as shown on the diagram.

We have to find the area of the trapezoidal region EDFG.

Formula to be used:

Area of a trapezium : 1/2 (sum of parallel sides) (height)

Or

Area of a trapezium : 1/2 (DE + FG) (BC)

We know that:

The tangent to a circle is perpendicular to the radius at the point of tang-ency.

So,

We have to find DH and FG.

In ΔOBE :

             ...Using Pythagoras formula.

In ΔOCF :

Plugging the values in the area equation.

⇒

⇒

⇒ square unit

The area of the trapezoidal region determined by the endpoints of the chords will be 99 square unit.

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.