1.An isosceles triangle with each leg measuring 13 cm is inscribed in a circle . if the altitude to the base is 12 cm find the radius of the circle

2. Circles a and b are tangent at point c. p is on circle a and q is on circle b such that pq is tangent to both circles. Given ac= 3 cm and bc= 8 cm, find pq

3.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

4. 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.

PLEASE ATLEAST ONE PLEASE THANKS

Sure, let's solve the first problem:

1. To find the radius of the circle, we can use the fact that the altitude of an isosceles triangle is also the perpendicular bisector of the base.

2. Since the given triangle is isosceles with each leg measuring 13 cm, we know that the base is also 13 cm.

3. The altitude to the base is given as 12 cm. This means that it divides the base into two equal parts, each measuring 6.5 cm.

4. Now, we have a right triangle formed by the radius of the circle, the altitude, and half of the base.

5. Using the Pythagorean theorem, we can calculate the radius of the circle as follows:
r^2 = h^2 + (b/2)^2
r^2 = 12^2 + 6.5^2
r^2 = 144 + 42.25
r^2 = 186.25
r ≈ √186.25
r ≈ 13.65 cm

Therefore, the radius of the circle is approximately 13.65 cm.

Let me know if you'd like me to solve another problem or if you have any further questions!