Question

in the figure below, ( ad = 6x - 33 ) and ( cd = 4x - 15 ). find the value of ( x ).

Explanation:

Step1: Recognize the property

Since \( \angle BAD = \angle BCD = 90^\circ \) and \( \angle ABD=\angle CBD = 34^\circ \), by the Angle - Angle - Side (AAS) congruence criterion, \( \triangle BAD\cong\triangle BCD \). In congruent triangles, corresponding sides are equal, so \( AD = CD \).

Step2: Set up the equation

We know that \( AD=6x - 33 \) and \( CD = 4x-15 \). Since \( AD = CD \), we can set up the equation:
\( 6x-33=4x - 15 \)

Step3: Solve for x

Subtract \( 4x \) from both sides of the equation:
\( 6x-4x-33=4x - 4x-15 \)
\( 2x-33=- 15 \)
Add 33 to both sides of the equation:
\( 2x-33 + 33=-15 + 33 \)
\( 2x=18 \)
Divide both sides by 2:
\( x=\frac{18}{2}=9 \)

Answer:

\( x = 9 \)