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# If in a $\Delta ABC$,$\angle A = 45^\circ$,$\angle C = 60^\circ$, then $a + c\sqrt 2$A. $b$B. $2b$C. $\sqrt {2b}$D. $\sqrt {3b}$

Last updated date: 22nd Mar 2023
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Answer
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Hint
In this case we are given the values of some angles of like $\angle A = {45^\circ },\angle C = {60^\circ }$ and asked to determine the value of $a + c\sqrt 2$ and we will use the extended sine rule to determine the relationship between the length of the triangle's sides and its circumradius to obtain the desired result.
Formula used:
Sine rule formula:
$\frac{a}{{sinA}} = \frac{b}{{sinB}} = \frac{c}{{sinC}} = 2R$
Complete step-by-step solution:
The given angle is $A = 45^\circ$, $C = 60^\circ$
$A + B + C = \pi$
By substituting the values on the equation, it becomes
$= > B = 75^\circ$
$a + c\sqrt 2 = k\sin A + k\sin C(\sqrt 2 )$
$= 2k(\frac{{\sqrt 3 + 1}}{{2\sqrt 2 }})$
The values on the equation becomes,
$= 2k\sin 75^\circ$
Then, the equation becomes
$= 2k\sin B$
$a + c\sqrt 2 = 2b$
So, option B is correct.
Note
You need to first determine the lengths of ABC in order to solve this problem. The lengths of AB and BC are $2$ and $3$, respectively. As a result, dragging a downward will result in the intersection of AD and BC.
Two lines are said to intersect when they have exactly one point in common. There is a point at which the intersecting lines meet. The point of intersection is the same location that appears on all intersecting lines. There will be a place where the two coplanar, non-parallel straight lines intersect. Here, point O, the intersection point, is where lines A and B meet.