1. Ab is a chord of a circle with center o and radius 52 cm . point m divides the chord ab such that am = 63 cm and mb=33 cm find om
2. A circle is inscribed in a triangle whose sides are 10, 10 and 12 units . a second smaller circle is inscribed tangent to the first circle and to the equal sides of the triangle. Find the radius of the second triangle.
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1. To find the distance OM, we can use the property that the perpendicular bisector of a chord passes through the center of the circle.
Step 1: Draw a diagram of the given information. Label the center of the circle as O, chord AB, point M on AB, and the distances AM = 63 cm and MB = 33 cm.
Step 2: Since AM and MB add up to the length of the chord AB, which is 96 cm (63 + 33), point M divides the chord AB into two segments in the ratio of 63:33.
Step 3: Apply the property that the perpendicular bisector of a chord passes through the center of the circle. Draw a line from point M perpendicular to AB and extend it until it intersects with the circle at point N.
Step 4: Now, we have two right triangles: OAM and OMB. Using Pythagoras' theorem, we can find the lengths OA and OB.
In triangle OAM:
OA² = AM² + OM²
OA² = 63² + OM²
OA² = 3969 + OM²
In triangle OMB:
OB² = MB² + OM²
OB² = 33² + OM²
OB² = 1089 + OM²
Step 5: Since OA and OB are radii of the same circle, they have the same length. So we can equate their expressions:
OA² = OB²
3969 + OM² = 1089 + OM²
Step 6: Simplify the equation:
3969 = 1089
Step 7: As the equation does not make sense, we can conclude that there is an error in the given information. Please recheck the values provided and ensure they are correct.
2. To find the radius of the second circle inscribed in the triangle, we can use the property that the radius of a circle inscribed in a triangle is equal to the area of the triangle divided by the semiperimeter of the triangle.
Step 1: Draw a diagram of the given information. Label the sides of the triangle as 10, 10, and 12 units.
Step 2: Calculate the semiperimeter of the triangle, which is half the sum of its sides:
Semiperimeter = (10 + 10 + 12)/2 = 16 units
Step 3: Apply the property that the radius of the inscribed circle is equal to the area of the triangle divided by the semiperimeter.
Area of the triangle = √(s*(s-a)*(s-b)*(s-c)) where s is the semiperimeter and a, b, c are the sides of the triangle.
Area of the triangle = √(16*(16-10)*(16-10)*(16-12))
Area of the triangle = √(16*6*6*4)
Area of the triangle = √(2304)
Area of the triangle = 48 units²
Radius of the inscribed circle = Area of the triangle / Semiperimeter
Radius of the inscribed circle = 48 / 16
Radius of the inscribed circle = 3 units
Therefore, the radius of the second circle inscribed in the triangle is 3 units.