Question

1. the graph of line a is shown. use the similar slope triangles to compare the slopes of segments gi and ik.
2. the graph of line t is shown. use the similar slope triangles to compare the slopes of segments bc and ce.
3. the graph of line t is shown. use the similar slope triangles to compare the slopes of segment ce and segment ac.
4. the graph of line u is shown. use the similar slope triangles to compare the slopes of segments gi and ik.

Explanation:

Problem 8:

Step1: Recall slope formula

The slope of a line segment between two points \((x_1,y_1)\) and \((x_2,y_2)\) is given by \(m = \frac{y_2 - y_1}{x_2 - x_1}\). For similar slope triangles, the ratio of rise (change in \(y\)) to run (change in \(x\)) should be equal if the segments are part of the same line.

Step2: Analyze segments \(GI\) and \(IK\)

• For segment \(GI\): Let's assume the grid has unit squares. From \(G\) to \(I\), let's say the change in \(y\) (rise) is \(y_G - y_I\) (negative since it's going down) and change in \(x\) (run) is \(x_I - x_G\) (positive).
• For segment \(IK\): From \(I\) to \(K\), change in \(y\) is \(y_K - y_I\) (negative) and change in \(x\) is \(x_K - x_I\) (0? Wait, no, looking at the graph, \(H\) and \(J\) are horizontal, \(G\) to \(H\) is horizontal, \(I\) to \(J\) is horizontal. So \(GI\) and \(IK\) are on the same line \(a\). So the slope of \(GI\) and \(IK\) should be equal because they are parts of the same straight line. The slope triangles for \(GI\) and \(IK\) are similar, so the ratio of rise over run is the same.

Step1: Slope formula

Slope \(m=\frac{\Delta y}{\Delta x}\), where \(\Delta y = y_2 - y_1\) and \(\Delta x = x_2 - x_1\). For a straight line, all segments on it have the same slope.

Step2: Analyze segments \(BC\) and \(CE\)

Segments \(BC\) and \(CE\) are parts of line \(t\). The slope triangles for \(BC\) and \(CE\) are similar (since the line is straight), so the ratio of rise to run for both segments will be equal. So the slope of \(BC\) is equal to the slope of \(CE\).

Step1: Slope formula

Slope \(m = \frac{\text{rise}}{\text{run}}=\frac{y_2 - y_1}{x_2 - x_1}\). For a straight line, all segments on it have the same slope.

Step2: Analyze segments \(CE\) and \(AC\)

Segments \(CE\) and \(AC\) are parts of line \(t\). The slope triangles formed by these segments are similar (as they lie on the same straight line), so the ratio of rise to run for \(CE\) and \(AC\) is the same. Thus, their slopes are equal.

Answer:

The slopes of segments \(GI\) and \(IK\) are equal.

Problem 9: