Question

1. what is the relationship among m∠bcd, m∠a, and m∠b in the figure below? 40° + 75° = m∠bcd 40° + 75° = 180° - m∠bcd 40° - 75° = m∠bcd 40° - 75° + m∠bcd = 180° 7. rectangle ghj k in the coordinate grid is dilated by a scale factor of 2 with the origin as the center of dilation, resulting in rectangle ghjk. what are the coordinates of j? (9, 8) (11, 6) (18, 4) (18, 8) 8. what is the slope of the line?

Explanation:

Step1: Recall exterior - angle property of a triangle

The measure of an exterior angle of a triangle is equal to the sum of the measures of the two non - adjacent interior angles. In \(\triangle ABC\), \(\angle BCD\) is an exterior angle, and \(\angle A = 40^{\circ}\), \(\angle B=75^{\circ}\). So \(m\angle BCD=m\angle A + m\angle B\), which means \(40^{\circ}+75^{\circ}=m\angle BCD\).

Step2: For dilation of a rectangle

If a point \((x,y)\) is dilated by a scale factor \(k\) with the origin as the center of dilation, the new coordinates \((x',y')\) are given by \((x',y')=(kx,ky)\). Assume the coordinates of point \(J\) are \((9,4)\) (by observing the grid). Since \(k = 2\), the new coordinates of \(J'\) are \((2\times9,2\times4)=(18,8)\).

Step3: Calculate the slope of a line

The slope \(m\) of a line is given by the formula \(m=\frac{\text{rise}}{\text{run}}\). By choosing two points on the line, say \((x_1,y_1)\) and \((x_2,y_2)\), if the rise (vertical change) is \(3\) and the run (horizontal change) is \(2\), then \(m=\frac{3}{2}\).

Answer:

1. \(40^{\circ}+75^{\circ}=m\angle BCD\)
2. D. \((18,8)\)
3. \(\frac{3}{2}\)