Video Transcript
If 𝑀𝐴𝐵𝐶 is a right triangular pyramid, the length of its lateral edge 𝑀𝐴 equals 59 centimeters, and its base 𝐴𝐵𝐶 is right angled at 𝐴, where 𝐵𝐴 equals 105 centimeters and 𝐶𝐴 equals 36 centimeters, find the height of the pyramid rounded to the nearest hundredth.
Let’s begin with a sketch of this pyramid. 𝐴𝐵𝐶 is the base with a right angle at 𝐴. We have two known side lengths on the base. 𝐶𝐴 is 36 centimeters and 𝐵𝐴 is 105 centimeters. We’re also given that the lateral edge 𝑀𝐴 is 59 centimeters. In this question, we are asked to work out the height or the perpendicular height of the pyramid, which will be this line in orange. If we define the height to be ℎ or ℎ centimeters, and then drawing this triangle given here in pink, then we can work out that we might be able to work out the perpendicular height using this right triangle. The problem is that we don’t yet have the length of the base of this pink triangle.
Let’s define this length to be 𝑥 centimeters and see how we could work out the value of 𝑥. Let’s have a closer look at what the base of this pyramid looks like. We have the two given lengths of 36 centimeters and 105 centimeters. And we know that there’s a right angle at 𝐴. The length of 𝑥 centimeters will be on the median from vertex 𝐴 to the other side, 𝐵𝐶. Remember that the centroid of a triangle is found by the intersection point of the three medians of the triangle. If we extended this line from vertex 𝐴 to the midpoint of 𝐵𝐶, then that would be the median from 𝐴.
So in order to work out the value of 𝑥, we’ll firstly need to work out the length of the median from vertex 𝐴 and then establish what proportion that the value of 𝑥 is of the median. In order to do this, we can use a very important geometric property. This property says that the median to the hypotenuse of a right triangle is half the length of the hypotenuse. In other words, this median from 𝐴 to the midpoint of 𝐵𝐶 is half of the hypotenuse which will be 𝐵𝐶. So if we know the length 𝐵𝐶, then we can calculate the length of the median. To find 𝐵𝐶, let’s apply the Pythagorean theorem.
The Pythagorean theorem states that in every right triangle, the square of the hypotenuse is equal to the sum of the squares on the other two sides. So in this triangle, we have the two sides 36 and 105. So we could square those and add them and that would be equal to 𝐵𝐶 squared. This is 1296 plus 11025 is equal to 𝐵𝐶 squared. Simplifying the left-hand side, we have 12321. Taking the square root of both sides, we have the square root of 12321 is equal to 𝐵𝐶. We then have that 𝐵𝐶 is equal to 111 centimeters. As we have now found the length of the hypotenuse in this triangle, we can return to the fact that the median to the hypotenuse is half of the length of the hypotenuse. That means that the length of the median from 𝐴 is half of 111 or 111 over two centimeters.
We now need to work out the value of 𝑥. We can do this by recalling the centroid theorem which tells us that the distance from each vertex to the centroid is two-thirds of the length of the median from this vertex. This means that 𝑥 is equal to two-thirds multiplied by 111 over two, which was the length of the median. Simplifying this, we have 111 over three, which is equal to 37. We have therefore worked out that the distance from the vertex 𝐴 to the centroid is 37 centimeters. This means that we now have enough information to work out the perpendicular height of the pyramid. We can then apply the Pythagorean theorem once more in this two-dimensional triangle.
This time, we have got the value of the hypotenuse 𝐶; it’s 59. The other two sides will be ℎ and 37. We therefore have ℎ squared plus 1369 is equal to 3481. We then subtract 1369 from both sides, leaving us with ℎ squared is equal to 2112. Taking the square root of both sides, we have that ℎ is equal to the square root of 2112 centimeters. Finally, as we’re asked for the answer to the nearest hundredth, we need to find the decimal equivalent. That will be 45.956 and so on. Rounding this to the nearest hundredth gives 45.96 centimeters.
Therefore, using the Pythagorean theorem and information about the centroid and the medians in a right triangle, we have found that the height of the pyramid is 45.96 centimeters to the nearest hundredth.
