MO circles

Juro built the ABCD square with a 12 cm side. In this square, he scattered a quarter circle that had a center at point B passing through point A and a semicircle l that had a center at the center of the BC side and passed point B. He would still build a circle that would lie inside the square and touch the quarter circle k, semicircle l and side AB. Find the radius of such circle.

Correct result:

r =  3 cm

Solution:

$$\begin{gathered} \mathrm{\mid }SO\mathrm{\mid }=\mathrm{\mid }SL\mathrm{\mid }+\mathrm{\mid }LO\mathrm{\mid }=R+6 \\ \mathrm{\mid }SB\mathrm{\mid }=\mathrm{\mid }BK\mathrm{\mid }-\mathrm{\mid }KS\mathrm{\mid }=12-r \\ \mathrm{\mid }OE\mathrm{\mid }=\mathrm{\mid }OB\mathrm{\mid }-\mathrm{\mid }BE\mathrm{\mid }=6-r \\ \mathrm{\mid }SE\mathrm{\mid }\text{²}=\mathrm{\mid }SO\mathrm{\mid }\text{²}-\mathrm{\mid }OE\mathrm{\mid }\text{²}=\mathrm{\mid }SB\mathrm{\mid }\text{²}-\mathrm{\mid }BE\mathrm{\mid }\text{² } \\ (6+r)\text{²}-(6-r)\text{²}=(12-r)\text{²}-r\text{² } \\ 12r+12r=144-24r \\ 48r=144 \\ r=144\mathrm{/}48=3\text{ cm} \end{gathered}$$

Try another example

Showing 0 comments:

Related math problems and questions:

• Triangle in a square
  In a square ABCD with side a = 6 cm, point E is the center of side AB and point F is the center of side BC. Calculate the size of all angles of the triangle DEF and the lengths of its sides.
• Square grid
  Square grid consists of a square with sides of length 1 cm. Draw in it at least three different patterns such that each had a content of 6 cm² and circumference 12 cm and that their sides is in square grid.
• Circle and square
  An ABCD square with a side length of 100 mm is given. Calculate the radius of the circle that passes through the vertices B, C and the center of the side AD.
• Mrak - cloud
  It is given segment AB of length 12 cm, where one side of the square MRAK laid on it. MRAK's side length 2 cm shown. MRAK gradually flips along the line segment AB the point R leaves a paper trail. Draw the whole track of point R until square can do the l
• Katy MO
  Kate draw triangle ABC. Middle of AB have mark as X and the center of the side AC as Y. On the side BC wants to find the point Z such that the content area of a 4gon AXZY was greatest. What part of the triangle ABC can maximally occupy 4-gon AXZY?
• Concentric circles and chord
  In a circle with a diameter d = 10 cm, a chord with a length of 6 cm is constructed. What radius have the concentric circle while touch this chord?
• Equilateral triangle ABC
  In the equilateral triangle ABC, K is the center of the AB side, the L point lies on one-third of the BC side near the point C, and the point M lies in the one-third of the side of the AC side closer to the point A. Find what part of the ABC triangle cont
• Concentric circles
  In the circle with diameter 19 cm is constructed chord 9 cm long. Calculate the radius of a concentric circle that touches this chord.
• Tangent 3
  In a circle with centre O radius is 4√5 cm. EC is the tangent to the circle at point D. Segment AB IS THE DIAMETER of given circle. POINT A is joined with POINT E and POINT B is joined with POINT C. Find DC if BC IS 8cm.
• ABCD square
  In the ABCD square, the X point lies on the diagonal AC. The length of the XC is three times the length of the AX segment. Point S is the center of the AB side. The length of the AB side is 1 cm. What is the length of the XS segment?
• Trapezoid MO-5-Z8
  ABCD is a trapezoid that lime segment CE is divided into a triangle and parallelogram, as shown. Point F is the midpoint of CE, DF line passes through the center of the segment BE, and the area of the triangle CDE is 3 cm². Determine the area of the trape
• Rhombus construction
  Construct ABCD rhombus if its diagonal AC=9 cm and side AB = 6 cm. Inscribe a circle in it touching all sides...
• Circles
  In the circle with a radius 7.5 cm are constructed two parallel chord whose lengths are 9 cm and 12 cm. Calculate the distance of these chords (if there are two possible solutions write both).
• MO SK/CZ Z9–I–3
  John had the ball that rolled into the pool, and it swam in the water. Its highest point was 2 cm above the surface. The diameter of the circle that marked the water level on the surface of the ball was 8 cm. Find the diameter of John ball.
• Square circles
  Calculate the length of the described and inscribed circle to the square ABCD with a side of 5cm.
• Isosceles - isosceles
  It is given a triangle ABC with sides /AB/ = 3 cm /BC/ = 10 cm, and the angle ABC = 120°. Draw all points X such that true that BCX triangle is an isosceles and triangle ABX is isosceles with the base AB.
• Two chords
  Calculate the length of chord AB and perpendicular chord BC to circle if AB is 4 cm from the center of the circle and BC 8 cm from the center of the circle.