Video Transcript
In a circle of center 𝑂, 𝐴𝐵 is
equal to 35 centimeters, 𝐶𝐵 is equal to 25 centimeters, and 𝐴𝐶 is equal to 40
centimeters. Given that line segment 𝑂𝐷 is
perpendicular to line segment 𝐵𝐶 and line segment 𝑂𝐸 is perpendicular to line
segment 𝐴𝐶, find the perimeter of triangle 𝐶𝐷𝐸.
We are given in the question the
length of the three sides of the triangle 𝐶𝐵𝐴. We know that 𝐴𝐵 is 35
centimeters, 𝐶𝐵 is 25 centimeters, and 𝐴𝐶 is 40 centimeters. We have been asked to calculate the
perimeter of triangle 𝐶𝐷𝐸. We will do this by firstly proving
that triangles 𝐶𝐵𝐴 and 𝐶𝐷𝐸 are similar using the chord bisector theorem. We notice from the diagram that the
line segments 𝑂𝐸 and 𝑂𝐷 both pass through 𝑂 and meet the chords 𝐴𝐶 and 𝐶𝐵
at right angles.
The chord bisector theorem states
that if we have a circle with center 𝑂 containing a chord 𝐵𝐶, then the straight
line that passes through 𝑂 and is perpendicular to 𝐵𝐶 also bisects 𝐵𝐶. In our diagram, this means that the
length of 𝐴𝐸 is equal to the length 𝐸𝐶 and the length 𝐶𝐷 is equal to the
length 𝐷𝐵.
It is also clear from the diagram
that 𝐴𝐶 is equal to two multiplied by 𝐸𝐶 and 𝐶𝐵 is equal to two multiplied by
𝐶𝐷. As the two triangles 𝐶𝐵𝐴 and
𝐶𝐷𝐸 also share the angle 𝐶, we have two corresponding sides in proportion and
the angle between the two sides is congruent. This proves that the two triangles
are similar. And in fact triangle 𝐶𝐵𝐴 is
larger than triangle 𝐶𝐷𝐸 by a scale factor of two, as the lengths of the
corresponding sides are twice as long. Side 𝐴𝐶 is equal to two
multiplied by side 𝐸𝐶, 𝐶𝐵 is equal to two 𝐶𝐷, and 𝐴𝐵 is equal to two
multiplied by 𝐸𝐷.
We can calculate the perimeter of
triangle 𝐶𝐵𝐴 by adding 40, 35, and 25. This is equal to 100
centimeters. The perimeter of triangle 𝐶𝐷𝐸
will therefore be equal to half of this. This is equal to 50
centimeters.
