# Video: Finding the Slopes of Lines and Using Them to Determine If the Lines Are Parallel

Consider the four points 𝐴 (1, 2), 𝐵 (6, 0), 𝐶 (−1, −2) and 𝐷 (3, −4). (i) Work out the slope of the line 𝐴𝐵. (ii) Work out the slope of the line 𝐶𝐷. (iii) Are the two lines parallel?

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### Video Transcript

Consider the four points 𝐴: one, two; 𝐵: six, zero; 𝐶: negative one, negative two; and 𝐷: three, negative four. Part one: Work out the slope of the line 𝐴𝐵. Part two: Work out the slope of the line 𝐶𝐷. Part three: Are the two lines parallel?

The slope of any line can be calculated by using the formula: 𝑦 two minus 𝑦 one divided by 𝑥 two minus 𝑥 one. This is the change in the 𝑦-coordinates divided by the change in the 𝑥-coordinates.

Let’s first consider the slope of 𝐴𝐵, where 𝐴 has coordinates one, two and 𝐵 has coordinates six, zero. Substituting these values into the formula gives us zero minus two divided by six minus one. Zero minus two is equal to negative two. And six minus one is equal to five. Therefore, the slope of the line 𝐴𝐵 is negative two-fifths.

Let’s now consider the slope of 𝐶𝐷, where 𝐶 has coordinates negative one, negative two and 𝐷 has coordinates three, negative four. Substituting these values into the formula gives us negative four minus negative two divided by three minus negative one. Minus four minus negative two is the same as minus four plus two. And minus four plus two is equal to negative two. On the denominator, three minus negative one is the same as three plus one. Three plus one is equal to four. Therefore, the slope of the line 𝐶𝐷 is minus two divided by four. This fraction can be simplified to give us the slope of the line 𝐶𝐷 as negative one-half.

The third part of our question asked if the two lines, 𝐴𝐵 and 𝐶𝐷, are parallel. Well, parallel lines have the same slope or gradient. So we need to consider whether negative two-fifths is the same as negative a half. Well, clearly these two fractions are different. Therefore, the lines 𝐴𝐵 and 𝐶𝐷 are not parallel to each other.