5.03 Right triangle applications in 3D
Much of what we have learned about finding the length of sides and measure of angles in triangles can be extended to shapes in three dimensions. The sine, cosine, and tangent ratios can be applied without modification to the faces of 3D shapes, and the 3D version of the Pythagorean theorem is very similar to the familiar 2D version.
The Pythagorean theorem
Consider a rectangular prism with side lengths $a$a m, $b$b m, and $c$c m. How can we determine the diagonal length of the prism? In the figure below we have labeled an intermediary length, $x$x m, which is the diagonal length of the bottom face.
We can use the Pythagorean theorem to find the value of $x$x as follows:
$x^2=a^2+b^2$x2=a2+b2.
If we now look at the triangle with side lengths $x$x m, $c$c m, and $d$d m, we can use the Pythagorean theorem again to relate these three lengths:
$d^2=x^2+c^2$d2=x2+c2.
But notice that this expression contains the term $x^2$x2, which we had previously found is equivalent to $a^2+b^2$a2+b2. So we can now write the length $d$d in terms of $a$a, $b$b, and $c$c only:
$d^2=a^2+b^2+c^2$d2=a2+b2+c2.
Looking back at the image of the prism we can see that each of the three side lengths correspond to segments that are perpendicular to the other two side lengths. Just as the Pythagorean theorem in 2D only works for right triangles, where there is a pair of perpendicular sides, so the corresponding theorem in 3D only works with three mutually perpendicular segments.
Given the three lengths $a$a, $b$b, and $c$c, of three mutually perpendicular sides of a rectangular prism, the length $d$d of the main diagonal is given by
$d=\sqrt{a^2+b^2+c^2}$d=√a2+b2+c2
The trigonometric ratios
Previously we have been using the sine, cosine, and tangent ratios to find properties of triangles that live in the 2D coordinate plane. Since the faces of 3D shapes are also flat 2D planes, we can apply these rules to triangular faces as well.
Practice questions
Question 1
A pyramid has a square base with side length $14$14 m and a vertical height of $24$24 m.
Find the length of the edge $\overline{AE}$AE. Round your answer to one decimal place.
If $\theta$θ is the measure of the surface angle $BEA$BEA, find $\theta$θ correct to the nearest minute.
Question 2
Two hot air balloons are moored to level ground below, each at a different location. An observer at each location measures the angle of elevation to the opposite balloon. The observers are $1600$1600 m apart.
Determine the difference in height between the two balloons. Round your answer to the nearest meter.
Use your result from part (a) to determine the distance separating the two balloons. Round your answer to the nearest meter.
Question 3
A square prism has sides of length $3$3cm, $3$3cm and $14$14cm as shown.
If the diagonal $HF$HF has a length of $z$z cm, calculate $z$z to two decimal places.
If the size of $\angle DFH$∠DFH is $\theta$θ°, find theta to two decimal places.