4.A nd B are two points in the space. Distances between A, B and a plane P are 8 cm and 4 cm, respectively. The distance between their intersection points with the plane P is 9 cm. Find the distance between A and B if;
a) A and B are in the same side of the plane.
b)A and B are in opposite sides of the plane.
5. In the adjacent figure, in plane α there is a circle and a line m tangent to the circle at a point T.
A is a point on circle such that PA ⊥ α where P is not in α.
If PT ⊥ m, PT = 10 cm, and PA = 8 cm, find the radius of the circle.
6.In the figure E is a plane, d is a line in E and the circle with center O is in E. From a point A outside of plane E we draw a perpendicular line AO to plane E. Given that, d is tangent to circle at B, BC = 8 cm, radius of circle is 6 cm and AO = 24 cm.
7.There is a circle in a plane. AB is the diameter of the circle. From point A we draw perpendicular line PA, to the plane. Given that PA = 5 cm and the radius of the cirle is 6 cm. Find the distance between the point P and the farrest point of the circle.
8.In the adjacent figure, PG is perpendicular to
the plane of equilateral triangle ABC at its centroid G.
If m(∠AFP) = 45°, what is
9.In the figure DDBC is an isosceles right triangle in the plane E. AD is perpendicular to plane E. Given that BD = 6 cm and DABC is an equilateral triangle. Find AD.
10.In the adjacent figure, PA ⊥ (ABC).
If m(∠APB) = m(∠APC) = 45° and
m(∠BPC) = 60°, find m(∠BAC).
11.Show that if two planes are perpendicular to the same line then these planes are parallel.
12.In the figure line CD is perpendicular to plane E. Given that AO = BO and C. Prove that DABC is an isosceles triangle.
13.State the following as true or false.
a.If a line is perpendicular to any line in a plane then it is perpendicular to the plane.
b.From any point of a line we can draw only one perpendicular to this line in space and in plane.
c. In plane P there are two lines d and l. All lines which are perpendicular to both of d and l are perpendicular to plane P.
14.In the figure, AH is perpendicular to plane E. d is a line in plane E and AB ^ d. Given that AB = 10 cm and AH = 7 cm. What is the smallest integer value of HC.
15.A triangle DABC is given in space. What is the set of all points which are equidistant from the vertices of the triangle.
16.In the figure PH E. Given that
m(<PAH) = 45°,
m(<PBH) = 50°,
m(<PCH) = 60°.
State the following as true or false.
a. PA > PB
b. PA < PC
c. PH = AH
d. PH > CH
e. PB < PC
f. AH > BH
g. HB > HC
17.In the figure PH is perpendicular to plane E. Given that PA > PB > PC. How do we order AH, BH and CH?
18.In the figure ABC is an isosceles triangle with CT = 7 cm, AB = 14 cm, AC = BC. Given that CD is perpendicular to the plane of ABC. If CD = 6 cm find DB.
19.In a plane there are two parallel lines d and k. A and B are two points on line d and AB = 6 cm. If the distance between these two lines is 5 cm find the area of DABC where C is a point on line k.
20.In the figure PT is perpendicular to plane E. ATB is a right triangle with AB = 17 cm. If PA = 25 cm and PT = 24 cm find BT.
21.In the figure ABCD is a rectangle with sides AB = 10 cm and BC = cm. Given that,
KD (ABCD) and
KD = 5 cm.
Find the area of DACK.
22.In the figure the line CD is perpendicular to plane M at point O. Given that OA = BO. Prove that DABC is isosceles triangle.
23.In the figure ABCD is a kite. We draw a line PD perpendicular to the plane of ABCD. Given that,
AB = 13 cm,
BK = 12 cm,
KD = 24 cm and
PD = 7 cm.
24.Two airplains with the same velocity are flying from the same plane at the same time. They make a 60° and 45° angles with the plane, respectively. When they collide they have 9000 m height. What was the distance between them initially?
25.In the figure AP and BQ are perpendicular to plane E. C is a point on PQ. Given that,