Video Transcript
In the figure below, 𝑀 is the
center of the circle, 𝑀𝐵 equals 15 centimeters, 𝐴𝐵 equals 20 centimeters, 𝑀𝐶
equals nine centimeters, and line 𝐴𝐵 is a tangent. Find the perimeter of the figure
𝐴𝐵𝐶𝑀.
We will begin by adding the
measurements to our diagram. We are told that 𝑀𝐵 is equal to
15 centimeters, 𝐴𝐵 is equal to 20 centimeters, and 𝑀𝐶 is equal to nine
centimeters. Our aim in this question is to find
the perimeter of the quadrilateral 𝐴𝐵𝐶𝑀. We already know the lengths of two
sides of this shape. We therefore need to calculate the
length of side 𝑀𝐴 and 𝐵𝐶. We will do this using the
properties of circles known as the circle theorems. Since the line 𝐴𝐵 is a tangent to
the circle, we know that triangle 𝑀𝐴𝐵 is a right triangle. This is because any tangent to a
circle is perpendicular to the radius at the point of contact.
As we know the lengths of two sides
of our right triangle, we can use the Pythagorean theorem to calculate the length
𝑀𝐴. This states that 𝑎 squared plus 𝑏
squared is equal to 𝑐 squared, where 𝑐 is the length of the hypotenuse, or longest
side, and 𝑎 and 𝑏 are the lengths of the other two sides. Substituting in our values, we have
15 squared plus 20 squared is equal to 𝑀𝐴 squared. The left-hand side simplifies to
225 plus 400, and this is equal to 625. We can then take the square root of
both sides of our equation. The square root of 625 is 25. And since 𝑀𝐴 is a length, the
hypotenuse of our triangle is 25 centimeters long.
An alternative method here would be
to recall the Pythagorean triples. One such example is a
three-four-five triangle. This means that any triangle with
side lengths in the ratio three, four, five will be a right triangle. In this triangle, the shortest
length was 15 centimeters. The next shortest length was 20
centimeters. Since three multiplied by five is
15 and four multiplied by five is 20, we could simply multiply five by five to
calculate the length of the hypotenuse. This once again confirms that the
length of side 𝑀𝐴 is 25 centimeters.
Let’s now consider triangle
𝑀𝐵𝐶. Once again, this is a right
triangle at 𝐶. We know this as a line from the
center of a circle that bisects a chord is also perpendicular to the chord. In this question, the chord 𝐵𝐷
has been bisected by the line from 𝑀 to 𝐶. This means that 𝑀𝐶 is the
perpendicular bisector of 𝐵𝐷. Once again, we have a right
triangle where we know the lengths of two sides. The hypotenuse is 15 centimeters,
and one of the shorter sides is nine centimeters. This is once again a
three-four-five triangle. Three multiplied by three is nine,
and five multiplied by three is 15. Multiplying four by three gives us
12. Therefore, the side 𝐵𝐶 has length
12 centimeters.
We could once again have calculated
this value using the Pythagorean theorem. We recall that we need to find the
perimeter of the quadrilateral 𝐴𝐵𝐶𝑀. This will be equal to the sum of
the lengths 𝐴𝐵, 𝐵𝐶, 𝐶𝑀, and 𝑀𝐴. We need to add 20, 12, nine, and
25. This is equal to 66. The perimeter of the figure
𝐴𝐵𝐶𝑀 is 66 centimeters.
