Question

1. (07.02 mc)

in \\(\delta bca\\), \\(cb = 11\text{ cm}\\), \\(cg = 6\text{ cm}\\), \\(ah = 9\text{ cm}\\). find the perimeter of \\(\delta bca\\). (1 point)

32 cm
40 cm
35 cm
42 cm

Explanation:

Identify given values and geometric properties

Using the Circle Geometry knowledge point

• Triangle \(\Delta BCA\) has an inscribed circle with center \(D\).
• The circle is tangent to the sides of the triangle at points \(F\), \(G\), and \(H\).
• Given lengths: \(CB = 11\text{ cm}\), \(CG = 6\text{ cm}\), and \(AH = 9\text{ cm}\).

Apply the Tangent Segments Theorem

Using the Circle Geometry knowledge point

• Tangent segments from an external point to a circle are equal in length:

$$ CF = CG = 6\text{ cm} $$

$$ AG = AH = 9\text{ cm} $$

$$ BF = BH $$

Calculate the remaining segment lengths

Using the Circle Geometry knowledge point

• Since \(CB = CF + FB = 11\text{ cm}\):

$$ FB = CB - CF = 11 - 6 = 5\text{ cm} $$

• Therefore, the equal tangent segment is:

$$ BH = FB = 5\text{ cm} $$

Calculate the perimeter of the triangle

Using the Circle Geometry knowledge point

• The perimeter \(P\) of \(\Delta BCA\) is the sum of its three sides:

$$ P = CB + BA + AC $$

$$ P = 11 + (BH + AH) + (CG + AG) $$

$$ P = 11 + (5 + 9) + (6 + 9) = 11 + 14 + 15 = 40\text{ cm} $$

Answer:

• (A) 32 cm
• (B) 40 cm (Correct answer)
• (C) 35 cm
• (D) 42 cm