# Question Video: Using Right-Angled Triangle Trigonometry to Solve Problems Involving Isosceles Triangles Mathematics • 10th Grade

π΄π΅πΆ is an isosceles triangle, where π΄π΅ = π΄πΆ = 10 cm, and π΅πΆ = 16 cm. Find the value of sin πΆπ΄π· given that π· is the midpoint of π΅πΆ.

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### Video Transcript

π΄π΅πΆ is an isosceles triangle, where π΄π΅ equals π΄πΆ, which equals 10 centimeters, and π΅πΆ equals 16 centimeters. Find the value of sin πΆπ΄π· given that π· is the midpoint of π΅πΆ.

Before we even try to find some trigonometric ratio, letβs begin by sketching out the triangle. We know that π΄π΅πΆ is isosceles, in other words, two sides of equal length. And the sides of equal length are π΄π΅ and π΄πΆ. These are shorter than the third side in the triangle π΅πΆ. And so perhaps triangle π΄π΅πΆ looks a little like this. Weβre then told that π· is the midpoint of π΅πΆ. Now, in fact, we know that if we bisect angle π΅π΄πΆ in our isosceles triangle, this angle bisector forms the line bisector of π΅πΆ as shown. So we can deduce that line π΄π· must be perpendicular to line π΅πΆ.

Now weβre trying to find the value of sin of πΆπ΄π·. So letβs begin by defining angle πΆπ΄π· to be equal to π. Then we can add this to our diagram as shown. Since π· is the midpoint of π΅πΆ, we can say that line segment π·πΆ must be equal to eight centimeters. Then we notice that triangle π΄π·πΆ is a right triangle for which we have an included angle weβ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 π. Now, of course, the trigonometric ratio for sin π is opposite divided by hypotenuse. So we will be able to find the value of sin πΆπ΄π· by dividing the opposite side to our included angle by the length of the hypotenuse.

Well, side π·πΆ sits directly opposite angle π΄. And the hypotenuse, the longest side of our triangle, lies opposite the right angle. So thatβs side π΄πΆ. In this case then, sin of π 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 πΆπ΄π· to be equal to π. So we can rewrite the left-hand side of this equation as sin of πΆπ΄π·. And that means, in fact, that weβre finished; weβve calculated the value of sin πΆπ΄π· given all the information in our question. Itβs equal to four-fifths.