Question

1. triangle mno and triangle pqr each have a side which lies on the line in the coordinate plane, as shown. • explain how the triangles are helpful in determining the slopes of mn and pq. • compare the slopes of mn and pq.

Explanation:

Step1: Recall slope formula

The slope formula is $m=\frac{\text{rise}}{\text{run}}$. The right - triangles formed with the line segments (MN and PQ) as hypotenuses have vertical sides representing the rise and horizontal sides representing the run.

Step2: Determine rise and run for each triangle

For the triangle with side MN, we can count the vertical and horizontal units between the endpoints of MN. Similarly, for the triangle with side PQ, we count the vertical and horizontal units between the endpoints of PQ. These values of rise and run are used in the slope formula.

Step3: Compare slopes

Since both MN and PQ lie on the same line, the ratio of rise to run (the slope) will be the same for both. The triangles are similar (by AA similarity as the angles formed with the x - axis are equal and the right - angles are equal), so $\text{rise}_1:\text{run}_1=\text{rise}_2:\text{run}_2$.

Answer:

The triangles are helpful because the vertical and horizontal sides of the right - triangles represent the rise and run values needed for the slope formula. The slopes of MN and PQ are equal because they lie on the same line, and the ratio of rise to run is the same for both line segments.