Question

In the given figure,find the value of \[x\] in terms of \[a\],\[b\] and \[c\].

Answer 1

Hint: We need to apply some triangle rule to find the value of \[x\] in terms of \[a\], \[b\] and \[c\]. First we will check if the triangles are similar or not . Then we have to find the corresponding angle in both triangles and apply the relation between the corresponding side . This is how we solve these kinds of problems.

Complete step by step answer:
Two polygons of the same number of sides are similar , if their corresponding angles are equal and their corresponding sides are in the same ratio.Here we can see two triangles - \[\Delta LMK\] and \[\Delta PNK\]. We have to prove they are similar . For that we can show their corresponding angles are equal or corresponding sides are in the same ratio. Here we can easily prove corresponding angles are equal.

In \[\Delta LMK\] the angle \[\angle LMK\]= \[{50^0}\] (from the picture we know that )
In \[\Delta PNK\] the angle \[\angle PNK\]= \[{50^0}\] (from the picture we know that )
So \[\angle LMK\]= \[\angle PNK\]
And \[\angle LKM = \angle PKN\] (common )
We know the triangle summation of three angles is \[{180^0}\]. So if two angles of some triangle are equal to two angles of another triangle then we can say that the third remaining angle of both the triangles is equal. (by the angle sum property)

Three angles of two triangles are equal so they can be called similar .(Angle-Angle-Angle similarity criteria )
\[\Delta LMK\] and \[\Delta PNK\] are similar triangles . According to a theorem if in two triangles , corresponding angles are equal , then their corresponding sides are in the same ratio and hence the two triangles are similar. Corresponding sides are \[LM\]and \[PN\].Another pair of corresponding sides are \[MK\] and \[NK\].
\[\dfrac{{PN}}{{LM}} = \dfrac{{NK}}{{MK}} = \dfrac{{KP}}{{KL}}\]
From the picture we know
\[LM = a\]; \[PN = x\] ; \[MK = \,MN + \,NK = b + c\]; \[NK\, = c\];
Putting these values in previous equation we get
\[\dfrac{x}{a} = \dfrac{c}{{b + c}}\];
\[\therefore x = \dfrac{{ac}}{{b + c}}\]

Hence, the value of x is $\dfrac{{ac}}{{b + c}}$.

Note: Be very careful about which triangles are taken and which angle and sides are corresponding angles . If we have to obtain the side value or have to show corresponding sides are in ratio then we will first prove the triangles are similar by Angle-Angle-Angle criteria and if we have to solve angle related problems then at first we must prove similarity between two triangles by corresponding side criteria.