Question

in the figure, ( mangle2=(d + 18)^{circ} ) and ( mangle3=(d + 22)^{circ} ). find the value of ( d ) such that ( overrightarrow{hl}perpoverrightarrow{hj} ).

Explanation:

Step1: Use the property of perpendicular lines

If \( \overrightarrow{HL}\perp\overrightarrow{HJ}\), then \(m\angle LHJ = 90^{\circ}\), so \(m\angle1 + m\angle2+m\angle3=90^{\circ}\).

Step2: Substitute the given angle expressions

Given \(m\angle2=(d + 18)^{\circ}\) and \(m\angle3=(d + 22)^{\circ}\), assume \(m\angle1 = 0^{\circ}\) (since no information about \(\angle1\) is given other than in the sum for the right - angle formed by \(HL\) and \(HJ\)). Then \((d + 18)+(d + 22)=90\).

Step3: Simplify the left - hand side of the equation

Combine like terms: \(d+d+18 + 22=90\), which simplifies to \(2d+40 = 90\).

Step4: Solve for \(d\)

Subtract 40 from both sides: \(2d=90 - 40\), so \(2d=50\). Then divide both sides by 2: \(d=\frac{50}{2}=25\).

Answer:

\(d = 25\)