Question

In $\Delta ABC$, $A - B - C,A - N - C$ and $\overline {MN} \left\| {\overline {BC} } \right.$, if $MN = 3$ and $BC = 7,$ that divides $\overline {AB} $ from A in ratio…….
A. $3:7$
B. $4:7$
C. $3:4$
D. $4:3$

Answer 1

Hint: According to given in the question we have to determine the ratio that divides $\overline {AB} $ from A. So, first of all we have to draw the triangle ABC and as mentioned in the question$\overline {MN} \left\| {\overline {BC} } \right.$, $MN = 3$ and $BC = 7,$ which is as given below:

Now, first of all we have to make the triangles ABC and AMN congruent with the help of two angles as we know that angle A is a common angle so, with the help of these angles we can easily make the both of the triangles ABC and AMN congruent.
Now, with the help of the congruent triangles ABC and AMN we can compare the sides of the triangle and then as we know that point M divides the line AB equally hence, we can easily determine the dimensions of the remaining sides

Complete step-by-step solution:
Step 1: First of all we have to make the triangles ABC and AMN congruent with the help of two angles as we know that angle A is a common angle so, with the help of these angles we can easily make the both of the triangles ABC and AMN congruent as mentioned in the solution hint. Hence,
$ \Rightarrow \angle BAC = MAN............(1)$
Step 2: Now, as we know that angle A is a common angle for both of the triangles ABC and AMN hence,
$ \Rightarrow \angle AMN = \angle ABC.........(2)$
Step 3: So, from the solution step 1 and 2 we have obtained (1) and (2) and with the help of (1) and (2) we can say that triangle ABC and AMN are congruent to each other.
$ \Rightarrow \Delta ABC \cong \Delta AMN.....................(3)$
Step 4: Now, as we have obtained that triangles ABC and AMN are congruent to each-other hence,
$ \Rightarrow \dfrac{{AM}}{{AB}} = \dfrac{{MN}}{{BC}}$…………………….(4)
But, as we know that $MN = 3$ and $BC = 7,$ so on substituting all the dimensions in the expression (4) as obtained just above,
$ \Rightarrow \dfrac{{MN}}{{BC}} = \dfrac{3}{7}$
$ \Rightarrow \dfrac{{AM}}{{AB}} = \dfrac{3}{7}$
Rearranging the terms in the fractions of the sides of the triangles as obtained just above,
$ \Rightarrow \dfrac{{AB}}{{AM}} = \dfrac{7}{3}$………………(5)
Step 5: Now, as we know that AB = AM + BM on substituting in the expression (5) as obtained in the solution step 4.
$
   \Rightarrow \dfrac{{AM + BM}}{{AM}} = \dfrac{7}{3} \\
   \Rightarrow \dfrac{{AM}}{{AM}} + \dfrac{{BM}}{{AM}} = \dfrac{7}{3}
 $
On substituting AM from the left side of the expression as obtained just above,
$
   \Rightarrow 1 + \dfrac{{BM}}{{AM}} = \dfrac{7}{3} \\
   \Rightarrow \dfrac{{BM}}{{AM}} = \dfrac{7}{3} - 1
 $
On determining the L.C.M of the expression as obtained just above,
$
   \Rightarrow \dfrac{{BM}}{{AM}} = \dfrac{{7 - 3}}{3} \\
   \Rightarrow \dfrac{{BM}}{{AM}} = \dfrac{4}{3}
 $
Hence,
$ \Rightarrow \dfrac{{AM}}{{BM}} = \dfrac{3}{4}$
Hence, we have obtained the required ratio for triangle ABC which is $\dfrac{{AM}}{{BM}} = \dfrac{3}{4}$.

Therefore option (C) is correct.

Note: If any one of the angles is a common angle for both of the two given triangles and one more angle is equal to the other angle then we can say that both of the two triangles are congruent to each-other.
If one angle and three sides of given two triangles or if two angles or one side of given triangles are equal to each-other then we can say that both of the triangles are congruent to each-other.