Q. 75.0( 1 Vote )

# State giving reasons, whether the following statements are true or false: In all the following questions the line does not contain a side of the triangle.

(1) A line can be drawn in the plane of a triangle not intersecting any of the sides of a triangle.

(2) A line can be drawn in the plane of a triangle which is not passing through any of the three vertices and intersecting all the three sides of the triangle.

(3) If a line drawn in the plane of a triangle intersects the triangle at only one point, the line passes through a vertex of the triangle.

(4) If a line intersects two of the three sides of a triangle in two distinct points and does not intersect the third side, then the line is parallel to the third side.

(5) In the plane of Δ ABC, a line / can be drawn such that = {P}, = {Q} and = ϕ.

Answer :

(1) The given statement is true because for a line in the plane of a Δ, there are 3 possibilities

i) The line does not intersect the Δ

ii) The line intersects the Δ at one point

iii) The line intersects the Δ at two points

The first possibility is satisfied; hence the given statement is true.

(2) The given statement is false because according to the theorem, if a line lying in the plane of a Δ and not passing through any vertex intersects one side, then it does intersect one more side but does not intersect the third side. Thus, a line not passing through a vertex of a Δ cannot intersect all the three sides.

(3) The given statement is true because a line intersecting a Δ and not passing through any vertex will intersect the Δ at two points.

(4) The given statement is false because if a line intersects two sides of a Δ and does not intersect the third side, it can intersect the line containing the third side.

(5) The given statement is true because in the plane of ΔABC, if line l intersects and at distinct points P and Q respectively, then according to the theorem, if a line lying in the plane of a Δ and not passing through any vertex intersects one side, then it does intersect one more side but does not intersect the third side, it cannot intersect.

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