Question

the segment \\( \overline { c d } \\) is tangent to \\( \odot t \\). find \\( m \angle t d c \\).

\\( m \angle t d c = \square \\)
(type an integer or a decimal. do not include the degree symbol in your answer.)

Explanation:

Step1: Use the property of tangent - radius

Since \( \overline{CD}\) is tangent to \( \odot T\) at \(C\), then \( \angle TCD = 90^{\circ}\). In \(\triangle TCD\), we know that the sum of the interior angles of a triangle is \(180^{\circ}\). So, \((8x + 1)+(4x + 5)+90=180\).

Step2: Solve the equation for \(x\)

Combine like terms: \(8x+4x+1 + 5+90=180\), which simplifies to \(12x+96 = 180\).
Subtract 96 from both sides: \(12x=180 - 96\), so \(12x=84\).
Divide both sides by 12: \(x=\frac{84}{12}=7\).

Step3: Find \(m\angle TDC\)

Substitute \(x = 7\) into the expression for \(m\angle TDC=(4x + 5)\).
\(m\angle TDC=4\times7+5=28 + 5=33\).

Answer:

33