Question

find the distance from point a to bd. round your answer to two decimal places.

Explanation:

Step1: Identify the formula for distance from a point to a line

The distance \( d \) from a point \( (x_0, y_0) \) to the line \( ax + by + c = 0 \) is given by \( d=\frac{\vert ax_0 + by_0 + c\vert}{\sqrt{a^2 + b^2}} \). First, find the equation of line \( BD \).

Step2: Find the slope of line \( BD \)

Points \( B(-2, -4) \) and \( D(8, 4) \). Slope \( m=\frac{4 - (-4)}{8 - (-2)}=\frac{8}{10}=\frac{4}{5} \).

Step3: Equation of line \( BD \)

Using point - slope form \( y - y_1=m(x - x_1) \) with \( B(-2,-4) \):
\( y+4=\frac{4}{5}(x + 2) \)
Multiply both sides by 5: \( 5y+20 = 4x + 8 \)
Rearrange to standard form: \( 4x-5y-12 = 0 \) (so \( a = 4 \), \( b=-5 \), \( c=-12 \))

Step4: Identify point \( A \)

Point \( A(3,-4) \), so \( x_0 = 3 \), \( y_0=-4 \)

Step5: Calculate the distance

Substitute into the distance formula:
\( d=\frac{\vert4\times3+(-5)\times(-4)-12\vert}{\sqrt{4^2+(-5)^2}}=\frac{\vert12 + 20-12\vert}{\sqrt{16 + 25}}=\frac{\vert20\vert}{\sqrt{41}}=\frac{20}{\sqrt{41}}\approx\frac{20}{6.4031}\approx3.12 \)

Answer:

\( 3.12 \)