Question

In the given figure PQ is parallel to MN. If $\dfrac{{KP}}{{PM}}$=$\dfrac{4}{{13}}$and $KN = 20.4cm$ Find KQ.

Answer 1

Hint: From the given question we have PQ is parallel to MN, therefore by using Thales theorem we can give the relation between the sides of the triangle. As it forms two triangles with equal angles and proportional sides. Based on this we can solve the equation to get the value of KQ.

Complete step by step solution:
Given: A triangle with vertices K, M, N.
And the line PQ is parallel to MN
In$\vartriangle {\text{KMN}}$, as PQ||MN, so,
$\dfrac{{{\text{KP}}}}{{{\text{PM}}}}$=$\dfrac{{{\text{KQ}}}}{{{\text{QN}}}}$(By Thales theorem)
Two triangles are similar when they have equal angles and proportional sides.

According to Thales Theorem, if we have three straight lines and they cut other two lines, then they produce proportional segments. When two triangles have a common angle and they have parallel opposite sides, we say they are in Thales position.
\[ \Rightarrow \dfrac{{{\text{KP}}}}{{{\text{PM}}}} = \dfrac{{{\text{KQ}}}}{{{\text{KN - KQ}}}}\]
\[ \Rightarrow \dfrac{4}{{13}} = \dfrac{{{\text{KQ}}}}{{{\text{20}}{\text{.4 - KQ}}}}\]
Cross multiplication
\[ \Rightarrow 4\left( {20.4 - {\text{KQ}}} \right) = 13{\text{KQ}}\]
\[ \Rightarrow {\text{KQ = }}\dfrac{{81.6}}{{17}}\]
${\text{KQ = 4}}{\text{.8 cm}}$
Therefore, the value of side KQ is 4.8 cm.
Thales theorem is also known as Basic proportionality theorem and can be defined as if a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points; the other two sides are divided in the same ratio.

Note: The mid-point theorem is a special case of the basic proportionality theorem. According to the mid-point theorem, a line drawn joining the mid-points of the two sides of a triangle are parallel to the third side. And we can write the converse of the theorem, that is if a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.