If the diagonals AC and BD intersect at point P in the cube in Figure what is the degree measure of angle APB ?

Options

A. 60
B. 65
C. 71
D. 83

Answer

C

Explanation

Look at the figure :

If we connect points A And D with a line segment and we connect points B and C with a line segment, then we will have a rectangle.

Side AB of rectangle ABCD is an edge of the cube. Since all edges of a cube have the same length, we can pick a number for the length of the edge of teh cube. Let's let the length of an edge of the cube be 1. So AB = 1. Diagonal AC of rectangle ABCD connects opposite vertices A and C of the cube. Now in any rectangular soild having a length l, a width w, and a height h, the formula for the distance d between opposite vertices is given by the formula `d = sqrt(l^(2)+w^(2)+h^(2))`. Here we have a cube, which is a rectangular solid where the length, width, and height are all equal. Since we are letting the edge of the cube be 1, we have that the length, width, and height are all equal to 1. So the length of AC is equal to `sqrt(1^(2)+1^(2)+1^(2))=sqrt(1+1+1)=sqrt(3)`.
We want the degree measure of angle APB. If we draw a perpendicular from point P to side AB of the rectangle and call the point where the perpendicular drawn from point P to side AB meets side AB point Q, then we will obtain two identical right triangles, triangles AQP and BQP.

Point Q bisects AB. So the length of `bar(AQ)=(1)/(2)`. Point P is the point of intersection of the diagonals, so point P is the midpoint of both diagonals. Therefore `AP=(AC)/(2)=(sqrt(3))/(2)`. Now angle APQ is `(1)/(2)` of angle APB, whose degree measure we are seeking. Let's find the degree measure of angle APQ. Then we will take twice this number for the degree measure of angle APB. If we refer to angle APQ by the letter `theta`, then we have that sin `theta=("opposite")/("hypotenuse")`
`=(AQ)/(AP)`
`=(((1)/(2)))/(((sqrt(3))/(2)))`
`=(1)/(2)xx(2)/(sqrt(3))`
`= (1)/(sqrt(3))`
Thus, `sin theta = (1)/(sqrt(3))`. So `theta` = arcsin `((1)/(sqrt(3)))`. Use your calculator to find that `(1)/(sqrt(3))~~0.57735` and that `theta` = arcsin `((1)/(sqrt(3)))~~35.26439^(@)`. So the degree measure of angle APB is approximately `2(35.26439)=70.52878`.