Question

1. points a, b, c, and d are collinear and positioned in that order. solve for x. find x if ab = 24, bc = 8, ad = 2x + 220, and cd = 4x + 310.

Explanation:

Step1: Determine the length of AC

Since A, B, C are collinear in order, \( AC = AB + BC \). Given \( AB = 24 \) and \( BC = 8 \), so \( AC = 24 + 8 = 32 \).

Step2: Express AD in terms of AC and CD

Since A, C, D are collinear in order (as A, B, C, D are collinear), \( AD = AC + CD \). Substitute \( AC = 32 \), \( AD = 2x + 220 \), and \( CD = 4x + 310 \) into the equation: \( 2x + 220 = 32 + 4x + 310 \).

Step3: Solve the equation for x

Simplify the right - hand side: \( 2x + 220 = 4x + 342 \).
Subtract \( 2x \) from both sides: \( 220 = 2x + 342 \).
Subtract 342 from both sides: \( 2x = 220 - 342=- 122 \).
Divide both sides by 2: \( x=-61 \).

Answer:

\( x = - 61 \)