Question

geometry topics 1-6 benchmark assessment

1. what is the equation of a line that is perpendicular to the line y = -2x + 3 and passes through the point (2, 4)?

text answer
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1. quadrilateral abcd is rotated 90° clockwise to produce abcd

select all of the true statements.
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□ ( overline{ab} cong overline{ab} )
□ if ( overline{ac} perp overline{bd} ), then ( overline{ac} perp overline{bd} )
□ ( mangle abc leq mangle abc )
□ ( mangle abc + mangle bcd = 180^circ )

Explanation:

Step1: Analyze the first problem (Equation of Perpendicular Line)

The given line is \( y = -2x + 3 \). The slope of a line perpendicular to a line with slope \( m \) is \( -\frac{1}{m} \). Here, the slope of the given line is \( -2 \), so the slope of the perpendicular line is \( \frac{1}{2} \).

Step2: Use Point - Slope Form

The line passes through \( (2, 4) \). Using the point - slope form \( y - y_1=m(x - x_1) \), where \( x_1 = 2 \), \( y_1 = 4 \) and \( m=\frac{1}{2} \), we have \( y - 4=\frac{1}{2}(x - 2) \).
Simplify: \( y-4=\frac{1}{2}x - 1 \), then \( y=\frac{1}{2}x+3 \).

Step1: Analyze the Second Problem (Dilation of Quadrilateral)

For \( \overline{AB}\cong\overline{A'B'} \)

Dilation is a similarity transformation. In a dilation, corresponding segments are proportional, not necessarily congruent (unless the scale factor is 1). But if we assume the dilation is a similarity transformation and we are just looking at the relationship of corresponding sides (if scale factor is 1 or not, but in general, for the property of dilation, if we consider the fact that in similar figures, corresponding sides are proportional, but if the problem is about the congruence of corresponding sides when dilated (maybe a specific dilation with scale factor 1? Or maybe the question is about the property of dilation preserving the shape, so corresponding sides are similar, but if we assume the dilation is an isometry (scale factor 1), then \( \overline{AB}\cong\overline{A'B'} \) is true.

For "If \( \overline{AC}\parallel\overline{BD} \), then \( \overline{A'C'}\parallel\overline{B'D'} \)"

Dilation preserves parallelism. So if two lines are parallel, their images under dilation are also parallel. So this statement is true.

For \( m\angle ABC\cong m\angle A'B'C' \)

Dilation preserves angle measures. So the measure of \( \angle ABC \) is equal to the measure of \( \angle A'B'C' \), so this statement is true.

For \( m\angle ABC + m\angle BCD=180^{\circ} \)

In a quadrilateral \( ABCD \), if \( ABCD \) is a parallelogram (since \( \overline{AC}\parallel\overline{BD} \) might imply it's a parallelogram), then consecutive angles are supplementary. Dilation preserves the angle measures, so \( m\angle ABC + m\angle BCD = 180^{\circ} \) holds for the original quadrilateral and its image. But let's check each statement:

• \( \overline{AB}\cong\overline{A'B'} \): True (if we consider the dilation as a transformation that preserves the length when scale factor is 1, or if the problem assumes that the dilation is such that corresponding sides are congruent. In general, for the property of dilation, if we are talking about the correspondence of sides in similar figures, but if the question is about the congruence, and if the dilation is an isometry, this is true.
• "If \( \overline{AC}\parallel\overline{BD} \), then \( \overline{A'C'}\parallel\overline{B'D'} \)": True (dilation preserves parallelism)
• \( m\angle ABC\cong m\angle A'B'C' \): True (dilation preserves angle measures)
• \( m\angle ABC + m\angle BCD = 180^{\circ} \): True (if \( ABCD \) is a parallelogram, consecutive angles are supplementary, and dilation preserves angle measures)

Answer:

For the first problem, the equation of the line is \( y=\frac{1}{2}x + 3 \). For the second problem, the true statements are \( \overline{AB}\cong\overline{A'B'} \), "If \( \overline{AC}\parallel\overline{BD} \), then \( \overline{A'C'}\parallel\overline{B'D'} \)", \( m\angle ABC\cong m\angle A'B'C' \), \( m\angle ABC + m\angle BCD = 180^{\circ} \) (depending on the properties of the quadrilateral \( ABCD \) and the dilation).