Question

given △abc with altitudes (overline{ad}), (overline{be}), and (overline{cf}) intersecting at orthocenter point g.
if ( mangle age = (12x + 10)^circ ) and ( mangle eag = (12x - 16)^circ ), determine the value of ( x ).
( x = square )

Explanation:

Step1: Use the property of right - angled triangle

In \(\triangle AEG\), since \(BE\perp AC\) (altitude), \(\angle AEG = 90^{\circ}\). By the angle - sum property of a triangle (\(\angle AEG+\angle EAG+\angle AGE=180^{\circ}\)), substituting \(\angle AEG = 90^{\circ}\), we get \(\angle EAG+\angle AGE=90^{\circ}\).

Step2: Substitute the given angle expressions

We know that \(m\angle AGE=(12x + 10)^{\circ}\) and \(m\angle EAG=(12x-16)^{\circ}\). Then \((12x - 16)+(12x + 10)=90\).

Step3: Simplify the equation

Combine like terms: \(12x+12x-16 + 10=90\), which is \(24x-6 = 90\).
Add 6 to both sides: \(24x=90 + 6\), so \(24x=96\).

Step4: Solve for \(x\)

Divide both sides by 24: \(x=\frac{96}{24}\).

Answer:

\(x = 4\)