Question Video: Determining Whether Two Lines are Perpendicular From Points on Each Line | Nagwa Question Video: Determining Whether Two Lines are Perpendicular From Points on Each Line | Nagwa

Question Video: Determining Whether Two Lines are Perpendicular From Points on Each Line Mathematics • Third Year of Preparatory School

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The line 𝐿₁ passes through the points (3, 3) and (βˆ’1, 0), and the line 𝐿₂ passes through the points (βˆ’3, 2) and (0, βˆ’2). Are the two lines perpendicular?

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Video Transcript

The line 𝐿 one passes through the points three, three and negative one, zero, and the line 𝐿 two passes through the points negative three, two and zero, negative two. Are the two lines perpendicular?

There are many ways of approaching this problem. One way is to begin by drawing a coordinate grid. We are told that the line 𝐿 one passes through the points three, three and negative one, zero as shown. The line 𝐿 two passes through the points negative three, two and zero, negative two. We are asked to work out whether the two lines are perpendicular. We know that two lines are perpendicular or meet at right angles if the product of their gradients or slopes is equal to negative one. In this question, we will find the gradient of line 𝐿 one, the gradient of line 𝐿 two and see if the product of these values equals negative one.

We know that the gradient or slope of any line is equal to the change in 𝑦 over the change in π‘₯. This can be written as 𝑦 one minus 𝑦 two over π‘₯ one minus π‘₯ two, where two points on the line have coordinates π‘₯ one, 𝑦 one and π‘₯ two, 𝑦 two. This is also sometimes known as the rise over the run. Let’s begin by considering line 𝐿 one. The gradient or slope of this line is equal to three minus zero over three minus negative one. Three minus zero is equal to three, and three minus negative one is equal to four as it is the same as three plus one. The gradient or slope of line 𝐿 one is equal to three over four or three-quarters.

We can repeat this for line 𝐿 two. The gradient here is equal to two minus negative two over negative three minus zero. This simplifies to four over negative three. Dividing a positive number by a negative number gives a negative answer. Therefore, the gradient or slope of line 𝐿 two is equal to negative four-thirds. We can now multiply our two values together. When multiplying two fractions, we multiply the numerators and denominators separately. As we’re multiplying a positive fraction by a negative fraction, we will get a negative answer. This is equal to negative 12 over 12, which simplifies to negative one. As the product of the two gradients or slopes is equal to negative one, we can conclude that the answer is yes, the two lines are perpendicular. Whilst it did look like this was the case from our diagram, it is important that we check using the coordinates.

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