Question

directions: use your resources to answer the following questions from unti 1: foundations of geometry. to best prepare you for the ia, ensure you are showing all annotations, steps, and work to receive full credit! #1.) in the diagram below \\( \overline { a c } \\) bisects \\( \overline { d e } \\) at \\( b \\). if \\( \overline { b c } = ( 5 x - 11 ) \mathrm { in } \\), \\( \overline { d b } = ( 4 x + 4 ) \mathrm { in } \\), \\( \overline { a b } = ( x + 17 ) \mathrm { in } \\), and \\( \overline { b e } = ( 8 x - 12 ) \mathrm { in } \\). find the length of \\( \overline { a c } \\). (a) 24 in. (b) 48 in. (c) 30 in. (d) 76 in. *#2.) the segment \\( a b \\) has endpoints \\( a ( - 10, - 5 ) \\) and \\( b ( 0, - 15 ) \\). determine half the length of \\( \overline { a b } \\). #3.) rita, \\( r \\), and zuri, \\( z \\), live at the locations below. if ayden, \\( a \\), lives equidistant between them, identify and label the coordinate point that represents adrian, \\( a \\). ***#4.) points \\( m, n \\), and \\( o \\) are on a line where \\( m \\) is the midpoint. if \\( \overline { n o } = 2 x ^ { 2 } - 16 \\) units and \\( m n = 2 x \\), determine the length of \\( \overline { m o } \\). (a) 8 units (b) 4 units (c) 32 units (d) 16 units

Explanation:

Step1: Use the property of bisector

Since \(\overline{AC}\) bisects \(\overline{DE}\) at \(B\), we have \(DB = BE\).
So, \(4x + 4=8x - 12\).

Step2: Solve the equation for \(x\)

Subtract \(4x\) from both sides: \(4 = 4x-12\).
Add \(12\) to both sides: \(16 = 4x\).
Divide both sides by \(4\): \(x = 4\).

Step3: Find the lengths of \(AB\) and \(BC\)

Substitute \(x = 4\) into \(AB=x + 17\), we get \(AB=4 + 17=21\) in.
Substitute \(x = 4\) into \(BC = 5x-11\), we get \(BC=5\times4-11=20 - 11 = 9\) in.

Step4: Calculate the length of \(AC\)

Since \(AC=AB + BC\), then \(AC=21+9 = 30\) in.

Answer:

C. 30 in.