Question

1. solve for bc in the picture to the right.

a) 57.0 b) 33.0 c) 43.4 d) 25.3
ac = 45, ab = 12

Explanation:

Step1: Identify Triangle Type

Triangle \( ABC \) is right - angled at \( B \) (since \( CB \) is a radius and \( AB \) is a tangent to the circle, so \( \angle ABC = 90^{\circ}\)).

Step2: Apply Pythagorean Theorem

The Pythagorean theorem states that for a right - triangle with hypotenuse \( c \) and legs \( a \) and \( b \), \( c^{2}=a^{2}+b^{2}\). In right - triangle \( ABC \), \( AC \) is the hypotenuse (\( AC = 45 \)), \( AB = 12 \) and \( BC \) is one of the legs. Let \( BC=x \). Then, by the Pythagorean theorem:

$$ AC^{2}=AB^{2}+BC^{2} $$

Substitute \( AC = 45 \) and \( AB = 12 \) into the formula:

$$ 45^{2}=12^{2}+x^{2} $$

$$ 2025 = 144+x^{2} $$

Step3: Solve for \( x \)

Subtract 144 from both sides of the equation:

$$ x^{2}=2025 - 144 $$

$$ x^{2}=1881 $$

Take the square root of both sides:

$$ x=\sqrt{1881}\approx43.4 $$

Answer:

c) 43.4