Video: Using Trigonometry to Find Lengths in Isosceles Triangles

Video Transcript

𝐴𝐵𝐶 is an isosceles triangle where 𝐴𝐵 is equal to 𝐴𝐶 which is equal to 10 centimeters and the angle 𝐶 is equal to 52 degrees 20 minutes and 21 seconds. Find the length of 𝐵𝐶 giving the answer to one decimal place.

In any question like this, it is always worth trying to draw a diagram first. We were told in the question that triangle 𝐴𝐵𝐶 is isosceles. This means that two of the sides have equal length. The length of 𝐴𝐵 and 𝐴𝐶 are both equal to 10 centimeters. We were also told that angle 𝐶 was equal to 52 degrees 20 minutes and 21 seconds.

Our first step is to convert this angle into just degrees. One degree is equal to 60 minutes. This means that we can change 20 minutes into degrees by dividing 20 by 60. 20 minutes is equal to 0.333 and so on degrees, or 0.3 recurring degrees. One degree is also equal to 3600 seconds, as there are 60 seconds in a minute. 60 multiplied by 60 is equal to 3600. We can, therefore, convert 21 seconds into degrees by dividing 21 by 3600. This is equal to 0.00583 and so on degrees.

We can now add these two values together to calculate 20 minutes and 21 seconds in degrees. This is equal to 0.33916 and so on degrees. We can, therefore, say that angle 𝐶 is equal to 52.34 degrees to two decimal places. Whilst we’ve rounded our answer here, it is important for accuracy to use the full answer on our calculator display in any further calculations.

Our next step is to create two identical right-angle triangles by drawing a vertical line from 𝐴 to the line 𝐵𝐶. We will call the point where this line meets 𝐵𝐶, 𝐷. We were asked to calculate the length of 𝐵𝐶. Well, the length of 𝐵𝐶 will be double the length of 𝐷𝐶, which we have labelled 𝑥.

We can use right angle trigonometry, or SOHCAHTOA, to calculate the length 𝑥. The length 𝐴𝐶 is the hypotenuse of the right-angled triangle, as it is the longest side. The length 𝐴𝐷 is the opposite, as it is opposite the 52.34-degree angle. Finally, the length 𝐶𝐷, labelled 𝑥, is the adjacent, as it is adjacent, or next to, the right angle and the 52.34-degree angle. We know the length of the hypotenuse. And we want to calculate the length of the adjacent. Therefore, we will use the cosine ratio.

This states that cos of 𝜃 is equal to the adjacent divided by the hypotenuse. Substituting in our values gives us cos of 52.34 is equal to 𝑥 divided by 10. Multiplying both sides of this equation by 10 gives us 𝑥 is equal to 10 multiplied by cos of 52.34. Typing this into the calculator gives us a value of 𝑥 of 6.109 and so on. We can, therefore, say that 𝐷𝐶 is equal to 6.109 and so on centimeters.

As previously mentioned, 𝐵𝐶 is double the length of 𝐷𝐶. Multiplying 6.109 and so on by two gives us 12.219 and so on. We were asked to give our answer to one decimal place. This means our answer needs to have one number after the decimal point. The deciding number is the one in the hundredths column. As this is less than five, we will round down. We can, therefore, say that the length 𝐵𝐶 in the isosceles triangle 𝐴𝐵𝐶 is 12.2 centimeters.