An isosceles triangle is inscribed in a circle. The shortest side is the base which is 16 cm long. If the radius of the circle is 10 cm, what is the length of side "a"?

I will assume "a" is one of the equal sides of the triangle

make a sketch, by drawing in the altitude to the base of the triangle.
draw in the radius to the base vertex.
You will have a right-angled triangle with sides 8 and x and hypotenuse 10
x^2 + 8^2 = 10^2
x = 6 ( you might have recognized the 3-4-5 right-angled triangle multiplied by a factor of 2 )

So the altitude is 6+10 = 16
Now you have a large right-angled triangle with sides 8 and 16 with hypotenuse "a"
a^2 = 16^2+8^2 = 320
a = √320 = 8√5

To find the length of side "a" in the isosceles triangle, we can use the property that in a circle, the radius drawn to the point of contact of a tangent is perpendicular to the tangent.

Here's how we can find the length of side "a":

1. Draw the isosceles triangle inscribed in the circle.
- The circle should have its center at the center of the triangle.
- Label the points where the sides of the triangle touch the circle as A, B, and C.
- Label the midpoint of the base (the shortest side) as D.

2. Draw the radius from the center of the circle to point D.
- Since D is the midpoint of the base, the radius drawn from the center of the circle to D is perpendicular to the base (side of length 16 cm).

3. Use the properties of an isosceles triangle.
- In an isosceles triangle, the base angles (the angles opposite the base) are equal.
- Therefore, angle ADB and angle ADC are equal.

4. Use the properties of a right triangle.
- Triangle ADB is a right triangle, with angle ADB equal to 90 degrees because the radius drawn from the center of the circle to D is perpendicular to the base AD.
- The hypotenuse of this right triangle is the radius of the circle, which is 10 cm.
- The shorter side of the right triangle is half of the base, which is 8 cm (half of 16 cm).

5. Use the Pythagorean theorem to find the length of side a.
- Apply the Pythagorean theorem: a^2 + 8^2 = 10^2.
- In this equation, "a" represents the length of side a.
- Solve for "a": a^2 + 64 = 100.
- Subtract 64 from both sides: a^2 = 36.
- Take the square root of both sides: a = √36 = 6 cm.

Therefore, the length of side "a" in the isosceles triangle is 6 cm.