Question Video: Finding the Length of a Side in an Isosceles Triangle given the Lengths of Its Legs and the Size of the Base Angles Mathematics

Video Transcript

𝐴𝐵𝐶 is an isosceles triangle where 𝐴𝐵 is equal to 𝐴𝐶 which is equal to 5.8 centimetres and angle 𝐵 is equal to 48 degrees eight minutes and 32 seconds. Find the length of 𝐵𝐶 giving the answer to one decimal place.

Where possible, it is always useful to draw a diagram. We were told that the triangle is isosceles, and that length 𝐴𝐵 was equal to length 𝐴𝐶. Both of these lengths were equal to 5.8 centimetres. We were also told the size of angle 𝐵, 48 degrees eight minutes and 32 seconds. We need to convert this into just degrees.

One degree is equal to 60 minutes. And one degree is also equal to 3600 seconds, as there are 60 seconds in a minute. And 60 multiplied by 60 is equal to 3600. We can, therefore, convert eight minutes into degrees by dividing eight by 60. This is equal to 0.1333 and so on degrees. We can convert 32 seconds into degrees by dividing 32 by 3600. This is equal to 0.00888 and so on degrees.

Adding these two numbers together will give us a value in degrees of eight minutes and 32 seconds. This is equal to 0.1422 and so on. We can, therefore, say that angle 𝐵 is equal to 48.14 degrees to two decimal places. Where possible, we want to avoid rounding our answers until the end to ensure that our final answer is as accurate as possible.

Our next step is to split the isosceles triangle 𝐴𝐵𝐶 into two right-angled triangles by drawing a vertical line from the vertex 𝐴 to the line 𝐶𝐵. We will call the foot of this vertical line 𝐷. We have been asked to work out the length of 𝐵𝐶. If we let the length of 𝐷𝐵 equal 𝑥, then we know that 𝐵𝐶 will be double this length, as the two right-angled triangles are identical.

We can now use right angle trigonometry, or SOHCAHTOA, to calculate the length 𝑥, which in turn will enable us to calculate the length of 𝐵𝐶. If we consider the triangle 𝐴𝐵𝐷, the length 𝐴𝐵 is the hypotenuse, as it is the longest side of the right-angled triangle. The length 𝐴𝐷 is the opposite, as it is opposite the angle we are working with. Finally, the length 𝐷𝐵 labelled 𝑥 is the adjacent, as it is adjacent, or next, to the 48.14-degree angle and the right angle.

We know the length of the hypotenuse. And we’re trying to calculate the length of the adjacent. Therefore, we will use the cosine ratio. This states that cos 𝜃 is equal to the adjacent divided by the hypotenuse. Substituting in our values gives us cos of 48.4 is equal to 𝑥 divided by 5.8. We can multiply both sides of this equation by 5.8. This gives us 𝑥 is equal to 5.8 multiplied by cos of 48.14. Typing this into the calculator gives a value for 𝑥 equal to 3.8702 and so on.

As 𝑥 was the length 𝐷𝐵, then 𝐷𝐵 is equal to 3.8702. We mentioned earlier that 𝐵𝐶 is double this length. Therefore, 𝐵𝐶 is equal to 7.7404 and so on. We were asked to give our answer to one decimal place. This means we need one number after the decimal point in the answer. The four in the hundredths column is the deciding number. As this is less than five, we will round down. This means that the length of 𝐵𝐶 to one decimal place is 7.7 centimetres.