Consider that circles 𝑀 and 𝑁 are
congruent and 𝐴𝐵 equals 𝐶𝐷. Find the value of 𝑥. Find the value of 𝑦.
Let’s start by seeing if we can
work out the value of 𝑥, which appears in the circle 𝑀. The line segments 𝑀𝐶 and 𝑀𝐷
will both be radii of the circle. And that means that they will be
the same length. This means that the triangle 𝑀𝐶𝐷
is an isosceles triangle. An isosceles triangle has two sides
of equal measure and two angles of equal measure. And so the measure of angle 𝑀𝐷𝐶
will equal the measure of angle 𝑀𝐶𝐷. They will both be 25 degrees.
We can then use the fact that the
interior angles in a triangle add up to 180 degrees to help us work out the value of
𝑥. So the three angles in the
triangle, 25 degrees, 25 degrees, and 𝑥 degrees, will add to give 180 degrees. We can then simplify and subtract
50 degrees from both sides of the equation, which gives us that 𝑥 degrees is equal
to 130 degrees. And so we’ve answered the first
part of the question. 𝑥 is equal to 130.
Now, let’s see how we calculate the
value of 𝑦 which appears in this circle 𝑁. The fact that these two circles are
congruent is really very important. Because the circles are congruent,
then the radii are also congruent. Therefore, we can say that the line
segments 𝐴𝑁 and 𝐵𝑁 are congruent, but these are also congruent to the line
segments 𝐶𝑀 and 𝐷𝑀. We are also told that 𝐴𝐵 is equal
to 𝐶𝐷. Now we know that there are two
equal chords in the circle which form part of these two triangles. Comparing these two triangles then,
we have three pairs of corresponding sides congruent. And so we can say that triangle
𝑀𝐶𝐷 is congruent to triangle 𝑁𝐵𝐴 by the SSS or side-side-side congruency
Notice that because these are
isosceles triangles, we could also have written that triangle 𝑀𝐶𝐷 is congruent to
triangle 𝑁𝐴𝐵. And because we want to work out
this angle measure of 𝑁𝐵𝐴, which is 𝑦 degrees, we can say that that’s equal to
the measure of angle 𝑀𝐶𝐷. And so 𝑦 degrees is equal to 25
degrees. And hence 𝑦 is equal to 25.
In this question, we proved that
two triangles are congruent. These triangles were each formed by
chords which are congruent and two radii. But in fact, this is a standard
property. It is worthwhile remembering the
fact that if chords of the same length connect radii in the same circle or in
congruent circles, then the two isosceles triangles formed are congruent. In this problem, we were given that
two chords were the same length which connected radii in congruent circles.