Question

geometry midterm

1. refer to the figure. select four points that are collinear.

figure with points a, b, c, d, e, f and a plane p
options: a, b, c, d, e, f (checkboxes)

1. fill in the blanks using the available answer choices.

in the diagram, \\(\overleftrightarrow{ab}\\) and \\(\overleftrightarrow{ec}\\) are perpendicular.
figure with points a, e, b on a horizontal line, c above e, h between e and b
if \\(m\angle heb = (7x)^\circ\\) and \\(m\angle ceh = (16x - 2)^\circ\\), then the value of x is \\(\underline{\quad}\\) (blank 1) and \\(m\angle heb = \underline{\quad}^\circ\\) (blank 2).
blank 1 options: 3, 4, 6
blank 2 options: 21, 28, 42

1. enter an equation in slope - intercept form for the line parallel to \\(y = 4x - 1\\) containing \\((-3, 2)\\).

Explanation:

Problem 2

Step1: Determine angle relationship

Since \(\overleftrightarrow{AB}\) and \(\overrightarrow{EC}\) are perpendicular, \(\angle CEB = 90^\circ\). So \(\angle HEB+\angle CEH = 90^\circ\). Substitute the given angle expressions: \(7x+(16x - 2)=90\).

Step2: Solve for \(x\)

Simplify the equation: \(23x-2 = 90\). Add 2 to both sides: \(23x=92\). Divide by 23: \(x = 4\).

Step3: Find \(m\angle HEB\)

Substitute \(x = 4\) into \(m\angle HEB=(7x)^\circ\): \(7\times4 = 28^\circ\).

Step1: Identify slope of parallel line

Parallel lines have the same slope. The line \(y = 4x-1\) has slope \(m = 4\), so the parallel line also has \(m = 4\).

Step2: Use point - slope form

Point - slope form is \(y - y_1=m(x - x_1)\). Substitute \(m = 4\), \(x_1=-3\), \(y_1 = 2\): \(y - 2=4(x + 3)\).

Step3: Convert to slope - intercept form

Expand: \(y - 2=4x+12\). Add 2 to both sides: \(y=4x + 14\).

Answer:

Blank 1: 4, Blank 2: 28

Problem 3