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.

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1. 1. Make a sketch by placing the triangle on the x-y gird so that the altitude falls along the y-axis with (0,12) the top of the triangle and (0,) the middle of the base.
Look at the righ-angled triangle in the first quadrant.
You know the hypotenuse is 13 and the height is 12, so by Pythagoras the base must be 5
So our circle must pass through A(0,12) and B(5,0) and the centre must lie on the y-axis or on x = 0

You also know that the centre must lie on the right bisector of AB
Slope of AB = -12/5
so slope of right bisector is 5/12
midpoint of AB is (5/2 , 6)
equation of right-bisector:
y = (5/12)x + b , with (5/2 , 6) on it, so
6 = (5/12)(5/2) + b
b = 6 - 25/24 = 119/24

so the centre is at P(0 , 119/24)
and the radius is 12 - 119/24 = 169/24
or appr 7.04

2. Please use capital letters for vertices
I assume the centres of your circles are A and B
Since the circles are tangents to each other, AB is straight line and AB = 11
Also you will have right angles at P and Q
From A draw a line parallel to PQ to hit BQ at D
APQD will be a rectangle.
You will have a right-angled triangle and you can find AD, thus PQ

Give the others a try, I will not be available for the rest of the afternoon

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2. equation 2x-3y=10 touches the circle with centre m (-2,4)

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