Question

a is equidistant to b and c. find bd. bd = ? b 11x + 9 d 4x + 30 c

Explanation:

Step1: Identify the property of isosceles triangle

Since \( A \) is equidistant to \( B \) and \( C \), triangle \( ABC \) is isosceles with \( AB = AC \). And \( AD \) is the altitude (perpendicular to \( BC \)), so by the property of isosceles triangle, \( D \) is the midpoint of \( BC \), which means \( BD = DC \).

Step2: Set up the equation

Given \( BD = 11x + 9 \) and \( DC = 4x + 30 \), we set them equal:

$$ 11x + 9 = 4x + 30 $$

Step3: Solve for \( x \)

Subtract \( 4x \) from both sides:

$$ 11x - 4x + 9 = 30 $$

$$ 7x + 9 = 30 $$

Subtract 9 from both sides:

$$ 7x = 30 - 9 $$

$$ 7x = 21 $$

Divide both sides by 7:

$$ x = \frac{21}{7} = 3 $$

Step4: Find \( BD \)

Substitute \( x = 3 \) into the expression for \( BD \):

$$ BD = 11(3) + 9 = 33 + 9 = 42 $$

Answer:

42