Question Video: Finding the Area of a Triangle Using the Euclidean Theorem Mathematics

Video Transcript

Calculate the area of triangle 𝐴𝐵𝐷.

We begin by recalling that the area of a triangle is half the length of the triangle’s base multiplied by the triangle’s perpendicular height. And since 𝐴𝐵𝐷 is a right triangle at 𝐷, we can choose 𝐵𝐷 as the base and then 𝐴𝐷 as the perpendicular height. The area of the triangle is then one over two times 𝐵𝐷 multiplied by 𝐴𝐷. We can find the length 𝐴𝐷 using the Euclidean theorem and then the length 𝐵𝐷 using the Pythagorean theorem.

If we first note that 𝐷 is the projection of 𝐵 onto the side 𝐴𝐶 and that the triangle 𝐴𝐵𝐶 is a right triangle at 𝐵, then the Euclidean theorem tells us that 𝐴𝐵 squared is equal to 𝐴𝐶 multiplied by 𝐴𝐷. Substituting now 𝐴𝐵 is 44 centimeters and 𝐴𝐶 is 55 centimeters, we have 44 squared is 55 multiplied by 𝐴𝐷. And dividing through by 55, we have 44 squared over 55 is 𝐴𝐷, which evaluates to 35.2 centimeters. And we now have the lengths of two sides of our right triangle 𝐴𝐵𝐷. And so we can use the Pythagorean theorem to find a third length, that is, the length 𝐵𝐷.

Applied to our triangle, the Pythagorean theorem tells us that 𝐴𝐵 squared is 𝐴𝐷 squared plus 𝐵𝐷 squared. And substituting 𝐴𝐵 is 44 centimeters and 𝐴𝐷 is 35.2 centimeters, we have 44 squared is 35.2 squared plus 𝐵𝐷 squared. And subtracting 35.2 squared from both sides and rearranging, this gives us 𝐵𝐷 squared is 44 squared minus 35.2 squared. Our right-hand side evaluates to 696.96. And taking the positive square root on both sides, since 𝐵𝐷 is a length, that’s the square root of 696.96, which is equal to 26.4. So 𝐵𝐷 is 26.4 centimeters.

So now, we can substitute 𝐴𝐷 is 35.2 centimeters and 𝐵𝐷 is 26.4 centimeters into our formula for the area. This gives us the area of triangle 𝐴𝐵𝐷 is one over two times 26.4 multiplied by 35.2. This evaluates to 464.64, and hence the area of triangle 𝐴𝐵𝐷 is 464.64 square centimeters.