Question Video: Using Right-Angled Triangle Trigonometry to Solve Problems Involving Isosceles Triangles Mathematics

Video Transcript

𝐴𝐵𝐶 is an isosceles triangle, where 𝐴𝐵 equals 𝐴𝐶, which equals 10 centimeters, and 𝐵𝐶 equals 16 centimeters. Find the value of sin 𝐶𝐴𝐷 given that 𝐷 is the midpoint of 𝐵𝐶.

Before we even try to find some trigonometric ratio, let’s begin by sketching out the triangle. We know that 𝐴𝐵𝐶 is isosceles, in other words, two sides of equal length. And the sides of equal length are 𝐴𝐵 and 𝐴𝐶. These are shorter than the third side in the triangle 𝐵𝐶. And so perhaps triangle 𝐴𝐵𝐶 looks a little like this. We’re then told that 𝐷 is the midpoint of 𝐵𝐶. Now, in fact, we know that if we bisect angle 𝐵𝐴𝐶 in our isosceles triangle, this angle bisector forms the line bisector of 𝐵𝐶 as shown. So we can deduce that line 𝐴𝐷 must be perpendicular to line 𝐵𝐶.

Now we’re trying to find the value of sin of 𝐶𝐴𝐷. So let’s begin by defining angle 𝐶𝐴𝐷 to be equal to 𝜃. Then we can add this to our diagram as shown. Since 𝐷 is the midpoint of 𝐵𝐶, we can say that line segment 𝐷𝐶 must be equal to eight centimeters. Then we notice that triangle 𝐴𝐷𝐶 is a right triangle for which we have an included angle we’re trying to find and we know two of the lengths. This means we can use right triangle trigonometry to find the value of sin of 𝜃. Now, of course, the trigonometric ratio for sin 𝜃 is opposite divided by hypotenuse. So we will be able to find the value of sin 𝐶𝐴𝐷 by dividing the opposite side to our included angle by the length of the hypotenuse.

Well, side 𝐷𝐶 sits directly opposite angle 𝐴. And the hypotenuse, the longest side of our triangle, lies opposite the right angle. So that’s side 𝐴𝐶. In this case then, sin of 𝜃 must be equal to eight-tenths. Simplifying eight-tenths by dividing both the numerator and denominator by two, and we see that eight-tenths is equivalent to four-fifths. But remember, we also defined angle 𝐶𝐴𝐷 to be equal to 𝜃. So we can rewrite the left-hand side of this equation as sin of 𝐶𝐴𝐷. And that means, in fact, that we’re finished; we’ve calculated the value of sin 𝐶𝐴𝐷 given all the information in our question. It’s equal to four-fifths.