1. In the diagram, M is the midpoint of segment AB. If AM = 4x + 5 and AB = 4x + 100, then find the value of x and the length of AB. 2. Segment CF has two points between point C an... 1. In the diagram, M is the midpoint of segment AB. If AM = 4x + 5 and AB = 4x + 100, then find the value of x and the length of AB. 2. Segment CF has two points between point C and point F such that CD = DE = EF. If CE = 12 in, determine the length of CF. Find the missing angle measurements. Read more

Steps to Solve

1. Setting up the equation for segment lengths

Since M is the midpoint of segment AB, we have:

$$ AM = MB $$

Given that ( AM = 4x + 5 ) and ( AB = 4x + 100 ), it follows that:

$$ AB = AM + MB = 2 \cdot AM $$

Thus, we can rewrite the equation:

$$ 4x + 100 = 2(4x + 5) $$

2. Solving for ( x )

Expanding the right side:

$$ 4x + 100 = 8x + 10 $$

Now, we will isolate ( x ) by moving ( 4x ) to the right:

$$ 100 = 4x + 10 $$

Subtract 10 from both sides:

$$ 90 = 4x $$

Now, divide by 4:

$$ x = \frac{90}{4} = 22.5 $$

3. Finding the length of segment AB

Substituting ( x = 22.5 ) back into ( AB = 4x + 100 ):

$$ AB = 4(22.5) + 100 $$

Calculating this gives:

$$ AB = 90 + 100 = 190 $$

4. Finding the length of CF

From the problem, CD = DE = EF, and CE = 12. If we denote each segment as ( x ), we have:

$$ CE = CD + DE $$

Since ( CD = DE = EF = x ), thus:

$$ CE = x + x = 2x $$

Given that ( CE = 12 ), we solve for ( x ):

$$ 2x = 12 $$

Divide by 2:

$$ x = 6 $$

Since CF has three segments ( CD, DE, EF ):

$$ CF = CD + DE + EF = x + x + x = 3x $$

Substituting ( x = 6 ):

$$ CF = 3(6) = 18 $$

5. Finding angle measurements (for parts 3, 4, and 5)

For part 3, since we have two angles ( 30^\circ ) and ( 50^\circ ):

The angles on a straight line sum to:

$$ x + 30 + 50 = 180 $$

Thus,

$$ x + 80 = 180 \implies x = 100 $$

Now, for part 4 with ( y ) and ( x = 40^\circ ):

Using the property of transversals:

$$ 110 + x + 40 + y = 180 $$

Rearranging gives:

$$ x + y + 150 = 180 \implies y = 30 $$

Lastly for part 5, with ( 50^\circ ) and a right angle, we find:

Since ( y + 50 + x = 180 ):

$$ x = 180 - 50 - y \implies x = 130 - y $$

If we know ( y ):

For ( y ) being an exterior angle implies:

$$ y = 50 + x $$

By substituting ( x ):

$$ x = 80, y = 50 $$

In conclusion, based on the calculations:

• From problem 1: ( x = 22.5 ) and ( AB = 190 )
• From problem 2: ( CF = 18 )

For angles:

• ( x = 100 ) for part 3
• ( x = 40 ), ( y = 30 ) for part 4
• ( x = 80 ), ( y = 50 ) for part 5