Find the area of triangle 𝐴𝐵𝐶, given that 𝐴𝐵 is equal to 𝐴𝐶, 𝐵𝐶 is equal to 20 centimeters, and cos of 𝐵 is equal to five over 13.
The area of a triangle is found by multiplying its base by its perpendicular height and dividing by two. In this question, we’ve only been given the length of one side of the triangle: 𝐵𝐶 is 20 centimeters. In order to work out the area, we also need to know the perpendicular height of this triangle which I’m going to refer to as 𝐴𝐷.
The question tells us that two sides of the triangle 𝐴𝐵 and 𝐴𝐶 are the same length. And therefore, triangle 𝐴𝐵𝐶 is isosceles. This means when I draw in a perpendicular height from the shared vertex of the two sides of equal length to the opposite side, this divides the triangle up into two identical right-angled triangles. This means that the length of 𝐵𝐶 20 centimeters is divided exactly in half into two lengths of 10 centimeters.
We still only know one length in each of these right-angled triangles. So let’s look at the other piece of information given in the question. We’re told that cos or cosine of the angle 𝐵 is equal to five over 13. Remember the definition of the cosine ratio in a right-angled triangle is that cos of a particular angle 𝜃 is equal to the length of the adjacent side divided by the length of the hypotenuse.
Let’s label the three sides of the right-angled triangle 𝐴𝐵𝐷 in relation to angle 𝐵. The hypotenuse and longest side of our right-angled triangle is the side directly opposite the right angle. So that’s side 𝐴𝐵. The opposite is the side opposite the known angle. So that’s side 𝐴𝐷. The adjacent is the final side. So that’s the side between the known angle and the right angle, in this case 𝐵𝐷.
Remember the cosine ratio tells us about the ratio between the adjacent and the hypotenuse. By substituting 10 for the length of the adjacent and 𝐴𝐵 for the hypotenuse, we know then that cos of 𝐵 is equal to 10 over 𝐴𝐵. This must also be equal to five over 13 as it’s stated in the question that cos of 𝐵 is equal to five over 13. This gives an equation that we can solve in order to find the length of 𝐴𝐵.
Now, ultimately, it isn’t 𝐴𝐵 that we want to calculate, it’s 𝐴𝐷, the perpendicular height of the triangle. But we aren’t in a position to calculate 𝐴𝐷 yet. However, if we can find 𝐴𝐵 first, we’ll then be able to calculate 𝐴𝐷 afterwards. Cross multiplying will eliminate the two denominators in this equation, giving 10 multiplied by 13 is equal to five multiplied by 𝐴𝐵. To find 𝐴𝐵, we need to divide both sides of the equation by five, giving 𝐴𝐵 is equal to 10 multiplied by 13 over five. A factor of five can be cancelled from both the numerator and denominator, giving two multiplied by 13 which is 26.
We found then that the length of the side 𝐴𝐵 is 26 centimeters. Remember ultimately, we’re looking to calculate the length of 𝐴𝐷. So let’s think about how we can do that now. We have a right-angled triangle, triangle 𝐴𝐵𝐷, in which we know the length of two of the sides. This means that we can apply the Pythagorean theorem in order to calculate the length of the third side.
The Pythagorean theorem tells us that in a right-angled triangle the sum of the squares of the two shorter sides is equal to the square of the hypotenuse. In our triangle, this means that 𝐴𝐷 squared plus 𝐵𝐷 squared is equal to 𝐴𝐵 squared. Substituting the known length of 𝐵𝐷 and 𝐴𝐵 gives 𝐴𝐷 squared plus 10 squared is equal to 26 squared.
And we now have an equation that we can solve in order to find the length of 𝐴𝐷. 10 squared is 100 and 26 squared is 676. So we have 𝐴𝐷 squared plus 100 is equal to 676. Subtracting 100 from each side of the equation, we have that 𝐴𝐷 squared is equal to 576. And then square rooting, we have that 𝐴𝐷 is equal to 24.
If you’re very familiar with Pythagorean triples — that is right-angled triangles in which all three sides are integers — you may have been able to spot this as 10, 24, 26 is an example of a Pythagorean triple. Whether you spotted it straightaway or whether you had to go through the working for the Pythagorean theorem, we now know that the perpendicular height of the triangle is 24 centimeters. And hence, we’re able to calculate the area.
Remember the base of the full triangle 𝐴𝐵𝐶 is 20 centimeters. So we multiply 20 by 24 and divide by two. A factor of two can be cancelled from both the numerator and the denominator, giving 10 multiplied by 24 over one which is just equal to 240.
The area of triangle 𝐴𝐵𝐶 is 240 centimeters squared.