FInd the measure of the segment

Question

􏰀M has radius 4.From external point E,a tangent is drawn meeting 􏰀M at T. Line ET has length 1/3 the length of line EM. Find ET.

Answer 1

Before we do anything to solve this, let's draw a picture to help visualize the problem. Draw a circle and label the center M. Make a point anywhere outside the circle and label it E. Draw a line connecting points M and E. Draw a line from point E tangent to the circle and label that point of intersection as T. Lastly, draw a radius from M to point T. You should now have triangle ETM.

We know MT is 4, so write a 4 along MT. We're also given that ET is 1/3 the length of EM. Label EM with x and label ET with x/3

A line tangent to a circle is perpendicular to the radius at that tangent point. In triangle ETM, point T is a right angle. We can now use the Pythagorean Theorem to solve for x.

a² + b² = c²

4² + (x/3)² = x²

16 + x²/9 = x²

144 + x² = 9x²

144 = 8x²

18 = x²

x = ±√18

x = √18 We're dealing with lengths that cannot be negative, so we ignore the negative answer.

x = 3√2

EM = 3√2

ET = x/3 = (3√2)/3 = √2