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.
Help me guys thnk you its very much appreciated
Sure! Let's solve these problems step by step.
1. To find the length of OM, we can use the Pythagorean theorem.
- The chord AB divides the diameter into two segments: AM and MB.
- To find the length of OM, we need to find the length of OB (radius) and subtract the length of BM from it.
- We know that AM = 63 cm, MB = 33 cm, and AB = AM + MB, so AB = 63 cm + 33 cm = 96 cm.
- Since AB is a chord passing through the center, it is twice the length of the radius, so OB = AB / 2 = 96 cm / 2 = 48 cm.
- Therefore, OM = OB - MB = 48 cm - 33 cm = 15 cm. So, the length of OM is 15 cm.
2. Let's denote the triangle's sides as A = 10 units, B = 10 units, and C = 12 units.
- The circle inscribed in the triangle is called the incenter, and it is the point where the angle bisectors of the triangle intersect.
- To find the radius of the second circle, we need to find the inradius of the triangle.
- The formula to find the inradius (r) of a triangle given its sides (A, B, and C) is r = (A + B - C) / 2.
- Substituting the values, we get r = (10 + 10 - 12) / 2 = 8 / 2 = 4 units.
- Therefore, the radius of the second circle is 4 units.
I hope this helps! Let me know if you have any further questions.