Question

circle c is inscribed in triangle jeg. what is the perimeter of triangle jeg? 60 units 65 units 70 units 75 units

Explanation:

Step1: Use the property of tangents from a common external point

Tangents from a common external point to a circle are equal. So, \(EK = EK = x + 5\), \(FG=GH = 2x\), \(JK=JH=x + 5\), \(GH = 3x-5\). Since \(FG = GH\), we have \(2x=3x - 5\).

Step2: Solve the equation for \(x\)

Subtract \(2x\) from both sides: \(0=x - 5\). Add \(5\) to both sides: \(x = 5\).

Step3: Calculate the side lengths

\(EG=EF+FG=(x + 5)+2x\), substitute \(x = 5\): \(EG=(5 + 5)+2\times5=10 + 10=20\). \(EJ=EK+KJ=(x + 5)+(x + 5)\), substitute \(x = 5\): \(EJ=(5 + 5)+(5 + 5)=20\). \(JG=JH+HG=(x + 5)+(3x - 5)\), substitute \(x = 5\): \(JG=(5 + 5)+(3\times5-5)=10+10 = 20\).

Step4: Calculate the perimeter

Perimeter of \(\triangle JEG=EG + EJ+JG\). Substitute the values: \(20+20 + 30=70\) (units).

Answer:

70 units