Question

. A plane bisects a 90° dihedral angle. From a point
on this plane 16 in. from the common edge,
perpendicular lines are constructed to the
respective faces of the dihedral angle. Find the
length of each perpendicular.

Answer 1

Let's call the point where the plane intersects the common edge of the dihedral angle point A. Let's also call the two faces of the angle that intersect at the common edge face 1 and face 2.

Since the plane bisects the angle, it divides the angle into two equal angles of 45 degrees each. Let's call the point where the perpendicular to face 1 intersects the plane point B, and the point where the perpendicular to face 2 intersects the plane point C.

We can draw a right triangle ABC, where AB and AC are the legs of the right triangle, and BC is the hypotenuse.

Since the angle between the perpendiculars and the plane is 45 degrees, we know that AB and AC are equal. Let's call this length x.

We also know that the distance from point A to the plane is 16 inches.

Using the Pythagorean theorem, we can relate the length of the hypotenuse to the length of the legs:

BC^2 = AB^2 + AC^2
BC^2 = x^2 + x^2
BC^2 = 2x^2
BC = sqrt(2x^2)

Now, we can use the fact that the distance from point A to the plane is 16 inches:

BC + 16 = length of perpendicular from point A to the dihedral angle

Substituting the expression we found for BC:

sqrt(2x^2) + 16 = length of perpendicular from point A to the dihedral angle

Simplifying this expression:

2x^2 + 32x + 256 = length of perpendicular from point A to the dihedral angle

Since AB = AC = x, the length of each perpendicular is x. Therefore, we have:

length of perpendicular to face 1 = length of perpendicular to face 2 = x

So, to find the length of each perpendicular, we need to solve for x:

2x^2 + 32x + 256 = length of perpendicular from point A to the dihedral angle

We can solve for x using the quadratic formula:

x = (-32 ± sqrt(32^2 - 4(2)(256 - length of perpendicular from point A to the dihedral angle))) / (2(2))

Simplifying:

x = (-16 ± sqrt(256 - length of perpendicular from point A to the dihedral angle)) / 2

Since x represents a length, we can eliminate the negative solution:

x = (-16 + sqrt(256 - length of perpendicular from point A to the dihedral angle)) / 2

Now we can substitute this expression for x into either of the expressions for the length of the perpendicular from point A to face 1 or face 2:

length of perpendicular to face 1 = length of perpendicular to face 2 = x
length of perpendicular to face 1 = length of perpendicular to face 2 = (-16 + sqrt(256 - length of perpendicular from point A to the dihedral angle)) / 2

Therefore, the length of each perpendicular is (-16 + sqrt(256 - length of perpendicular from point A to the dihedral angle)) / 2 inches.