Video Transcript
Given that the line segment π΄π΅ is tangent to the circle with center π at π΄, π΄π
equals 8.6 centimeters, and ππ΅ equals 12.3 centimeters, find the length of the
line segment π΄π΅, and round the result to the nearest tenth.
Letβs begin by adding the information given in the question onto the diagram. π΄π is 8.6 centimeters long. ππ΅ is 12.3 centimeters. And the length weβre looking to find is the length of the line segment π΄π΅. Now we notice that we have a triangle, triangle π΄ππ΅, for which we know the lengths
of two of the sides. Our first thought then may be that we can apply the Pythagorean theorem. But remember the Pythagorean theorem is only valid in right triangles. So we need to consider whether the triangle π΄ππ΅ is a right triangle or not.
The other key piece of information given in the question is that the line segment
π΄π΅ is tangent to the circle at the point π΄. A key property about tangents to circles is that a tangent to a circle is
perpendicular to the radius of the circle at the point of contact. So the line segment π΄π΅ is perpendicular to the radius π΄π. And we therefore have a right angle at π΄ in our triangle π΄ππ΅. And so the measure of angle ππ΄π΅ is 90 degrees, and we do indeed have a right
triangle. So we can apply the Pythagorean theorem to calculate the length of the third
side.
The Pythagorean theorem tells us that in a right triangle, the sum of the squares of
the two shorter sides is equal to the square of the longest side of the
triangle. Remember, the longest side, or hypotenuse, is always the side directly opposite the
right angle. So, in this case, thatβs the side ππ΅, which is 12.3 centimeters long. Substituting π΄π΅ and 8.6 for the two shorter sides of the triangle and 12.3 for the
longest side, or hypotenuse, we have the equation π΄π΅ squared plus 8.6 squared is
equal to 12.3 squared. We can evaluate 8.6 squared and 12.3 squared and then subtract 73.96, thatβs 8.6
squared, from both sides, giving π΄π΅ squared equals 77.33.
We solve this equation by taking the square root on both sides. And weβre only going to take the positive value here as π΄π΅ is the length of a side
of a triangle, which is a positive quantity. Evaluating this square root on a calculator, we find that π΄π΅ is equal to 8.79374
and so on. Remember though that weβre asked to round the result to the nearest tenth, which is
to one decimal place. And as thereβs a nine in the hundredths column, we round up, giving π΄π΅ equal to 8.8
centimeters.