In the triangle $A B C$ , $B D$ bisects angle $B$ and is perpendicular to $A C$ . If the lengths of the sides of the triangle are expressed in terms of x and y as shown, find the values of x

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Solution

The correct option is D 4
In $△$ s ABD and CBD,
$∠ A B D = ∠ C B D$ (Given)
$B D = B D$ (Common side)
$∠ A D B = ∠ C D B = 90^{\circ}$ (Given)
Thus, $△ A B D ≅ △ C B D$
Hence, $A B = B C$ and $A D = C D$ (Corresponding sides of congruent triangles)
$3 x + 1 = 5 y - 2$ and $x + 1 = y + 2$
$3 x = 5 y - 3$ (1) and $x = y + 1$ (2)
Put $x = y + 1$ in equation (1)
$3 (y + 1) = 5 y - 3$
$3 y + 3 = 5 y - 3$
$2 y = 6$
$y = 3$
Hence, $x = 3 + 1 = 4$

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