Video: Finding Slopes of Straight Lines

Suppose that points 𝐴(βˆ’3, βˆ’1), 𝐡(1, 2), and 𝐢(7, 𝑦) form a right-angled triangle at 𝐡. What is the value of 𝑦?


Video Transcript

Suppose that the points 𝐴: negative three, negative one; 𝐡: one, two; and 𝐢: seven, 𝑦 form a right-angled triangle at 𝐡. What is the value of 𝑦?

We can go ahead and make a sketch of these points. 𝐴 is negative three, negative one. 𝐡 is one, two. We know the π‘₯-coordinate of point 𝐢 is seven. And that means 𝐢 will be located somewhere along this line. We know the line 𝐴𝐡. And we’ve been told that the right angle of this triangle is at point 𝐡. We can get a general idea of where we think point 𝐢 would be. But this is not a good way to find an accurate answer. But because we know this is a right triangle, we could say that line 𝐴𝐡 is perpendicular to line 𝐡𝐢. That means the slope of line segment 𝐡𝐢 is the negative reciprocal of the slope of line segment 𝐴𝐡.

To solve this problem, we’ll need to do three things. First, find the slope of line segment 𝐴𝐡. Use that slope to find the negative reciprocal, which is the slope of line segment 𝐡𝐢. Then, take the slope of line 𝐡𝐢 and use that to find the 𝑦-value in point 𝐢. If we have two points, we find the slope by using π‘š equals 𝑦 two minus 𝑦 one over π‘₯ two minus π‘₯ one. For the points 𝐴 and 𝐡, that would be two minus negative one over one minus negative three, which equals three-fourths. The slope of line 𝐴𝐡 is then three-fourths. And we’ve completed step one.

For step two, we need to take the negative reciprocal of the slope we found in step one. The negative reciprocal of three-fourths is negative four-thirds. And that is step two. Now, for step three, we’ll take point 𝐡: one, two and point 𝐢: seven, 𝑦. We’ll let 𝐡 be π‘₯ one, 𝑦 one and 𝐢 be π‘₯ two, 𝑦 two. The slope negative four-thirds is equal to 𝑦 minus two over seven minus one. Seven minus one is six. To solve this, we cross multiply. Negative four times six equals three times 𝑦 minus two. Negative 24 equals three 𝑦 minus six.

To give us a bit more room to solve for 𝑦, we add six to both sides and we get negative 18 equals three 𝑦. Divide both sides of the equation by three and we get negative six equals 𝑦. And we found from step three that 𝑦 must be equal to negative six. This means, for this to be a right triangle, point 𝐢 needs to be located at seven, negative six. And so, we found that missing value to be negative six.

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