B. Using Projection in Calculations

12. In the figure ABCD and BCLK are two congruent squares.

Given that AB BK and DL = 8 cm.

Find DK.

13. In the plane α, consider the points B, C and E so that BC = 9 cm, CE = 12 cm and BE = 15 cm. Let A be a point not in plane α such that AB ⊥ α and AB = 36 cm.

a) Find the angle between line CE and plane ABC.

b) Find the tangent of the angle between planes ACE and BCE.

14. In the figure E₁ and E₂ are two intersecting planes and their intersection is the line BC. The points K and P are on the planes E₁ and E₂, respectively. Given that,

KC BC,

PC BC,

KC = PC = cm and

KP = 9 cm.

Find the angle between the planes E₁ and E₂.

15. In the figure B, C, D are in the plane E. The propejction of point A in the plane E is D. Given that,

BD DC,

BD = DC = 6 cm and

AD = 8 cm.

Find the sine of the dihedral angle between the plane E and the plane (ABC).

16*. Given ABC and ABD two equilateral triangles included in planes α and β respectively. If α ⊥ β, calculate the angle between lines AB and CD.

17*. In triangle ABC we have BC = 10 cm and

cos B ⋅ cos C = cos A + sin B ⋅ sin C.

Given a point O not in the plane of triangle ABC with the property OA = OB = OC = 13 cm.

Find the cosine of the angle between line OA and plane ABC.

18. In the figure the dihedral angle between P and Q is 120. The intersection of the planes is line d. Line k is equidistant from both of the planes and 10 cm away from the line d. Find the distance between k and plane Q.

19. The dihedral angle between P and Q is 60. There is a circe in plane P with radius 4 cm. What is the area of the projection of this circle in plane Q?

20. In the figure ABCD is a rhombus. PA is perpendicular to the plane of ABCD. The distance from A to BC is 4 cm and DC = 5 cm. If PA = cm find the area of DPAC.

21. In the figure two planes are parallel to each other.

The line d make 60° angle with these planes. If the distance between the intersection points of line d and the planes is 12 cm find the distance between these two planes.

22. In the figure DABC is an equilateral triangle and DBDC is an isosceles right triangle. Given that DC = cm

and dihedral angle between the planes of (ABC) and (BDC) is 150°. Find AD.

23. In the figure BCDE is a rectangle and DABC is an isosceles triangle. The projection of point A is on the intersection point of the diagonals of BCDE. Given that,

BC = 30 cm and the projection of AK on plane (BCDE) is 20 cm. Find the sum of the lengths of diagonals of the rectangle.

24. Decide whether a trihedral angle can be constructed in which the face angles are respectively;

a) 70°, 80°, and 100°

b) 25°, 30°, and 75°

c) 70°, 115°, and 190°

d) 120°, 140°, and 160°

25. Can a trihedral angle be constructed with the following dihedral angles?

a) 130°, 25°, 75°

b) 30°, 105°, 35°

c) 60°, 80°, 100°

d) 60°, 190°, 140°

26*. A right triangle ABC is projected onto a plane α which is parallel to BC and passes through vertex A, so that projections AB' and AC' of sides AB and AC have the lengths 3 cm and 5 cm and cos ∠B'AC' = . Calculate the distance from point B to plane α.

27. In the figure line AB cut the plane P by a 30° angle. If the length of AB is 20 cm find the length of its projection on the plane P.

28. In the figure P is a plane and it is passing through the center of given circle. The circle is inclined for 45°. The area of the projection of the circle is cm².

Find the radius of the circle.

29. Area of a rectangle in a plane P is a. If the area of the projection of this rectangle on another plane Q is find the angle between these two planes.

30. In the figure ABCD is a rectangle and PCD is a triangle. Point T is the center of rectangle and the projection of point P in the plane of (ABCD). The dihedral angle between the planes of ABCD and PCD is 60°.

Given that A(DPCD) = 10 cm².

Find A(ABCD).

31. In the adjacent figure, a cube is given with one side is m units.

Find distance between

a) B1D and D1C1.

b) BC1 and AC.

c) B1C and BD1. (in terms of m)

32. Given a right trapezoid ABCD where ∠A = ∠B = 90°. Line segment KA is perpendicular to the plane of ABCD. AD = b and AB = a with ∠C = α.

Find the distance between

a) AK and BC

b) KD and BC

c) AK and CD

33. In the adjacent figure, ABC is an equilateral triangle with a side of 6 cm. A₁B₁C₁ is the

corresponding projection of ΔABC on a parallel plane. AA₁= BB₁ = CC₁ = 6 cm and they are perpendicular to plane ABC. Find the distance

between AA₁ and B₁C.

34. Through vertex B of ΔABC drawn a line c, perpendicular to the plane of ΔABC. Find the distance between line c and AC if AC = 25 cm, BC = 15 cm, and ∠ABC = 90°.

35. Given a square ABCD and O the intersection point of its diagonals. Line segment MO is perpendicular to the plane of the square and MO = units. If one side of the square is 2a units, find:

a) the distance between AB and MO

b) the distance between BD and MC in terms of a.

36. In a plane α given a circle with center O and radius r. Through point C on the circle drawn a line c, perpendicular to plane α. Line d, lying in plane α, tangent to the given circle at point A. What is the distance between lines d and c if ∠AOC = 120° in terms of r?

37. Through vertex A of square ABCD with side length m drawn a plane α perpendicular to AC. Find the distance between BD and a line c which is lying in α and not parallel to BD, in terms of m.

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