A. Types of Projection

1. DABC is a triangle in a plane and DA₁B₁C₁ is its projection in another plane. Given that AA₁ = 6 cm, BB₁ = 9 cm and CC₁ = 12 cm. Find the distance between the centroids of these triangles.

2. Two intersecting planes α and b are given. Line d is their intersection. Given that a line n is perpendicular to α and line m is parallel to n. What can we say about m and d?

3. An equilateral triangle DABC is given in a plane α.

a. Can we always say that its projection is also an equilateral triangle?

b. If plane b is passing through AB then can we say that projection of DABC on b is equilateral?

c. At which condition the projection of DABC in plane b is an equilateral triangle?

4. ABCD is a square in plane α. Plane b passes through AB. The distance between the intersection point of diagonals of ABCD and plane b is 7 cm. Find the distance between point D and plane b.

5. Given that AB and CD are two line segments in space. Both of AB and CD are parallel to line d. Distance between AB and d is 12 cm, CD and d is 6 cm. If both of the lengths of AB and CD are 20 cm and length of projection of AB in line d is 10 cm find the length of the projection of line CD in d?

6. An isosceles right triangle has legs 12 cm. The hypothenuse of this triangle is parallel to a line d. What is the length of the projection of this triangle in line d?

7. Can we say that the projection of a right angle in a plane is also a right angle if none of the arms of the angle are parallel to given plane?

8. In the figure the lines d₁ and d₂ are perpendicular to each other. The projection of point S on the plane of lines d₁ and d₂ is point A. The points K, L, M and N are equidistant from A. Given that,

KM = 24 cm and KS = 13 cm.

Find A(DKNS) + A(DKMS).

9. In the figure G is centroid of equailateral triangle DABC. R is an exterior point and RG (ABC). Given that,

AB = 12 cm and

RG = 4 cm.

Find RC.

10. In the figure the planes P and Q intersected alond the line d. Point A is in plane P and AB d. T is the projection of point A in plane Q. Given that,

AT = 4 cm,

TB = 3 cm and

BC = 5 cm.

Find AC.

11. In the figure the lines d₁, d₂, d₃ are perpendicular to the parallel planes P and T. The three lines are equidistant from each other. Distance between the planes P and T is 16 cm.

If R is the midpoint of AK and AK = find the distance from S to R.

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