Question Video: Using the Properties of Parallel Lines and Solving Linear Equations to Determine the Value of an Unknown and the Length of a Line Segment | Nagwa Question Video: Using the Properties of Parallel Lines and Solving Linear Equations to Determine the Value of an Unknown and the Length of a Line Segment | Nagwa

# Question Video: Using the Properties of Parallel Lines and Solving Linear Equations to Determine the Value of an Unknown and the Length of a Line Segment Mathematics • First Year of Secondary School

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In the diagram below, π΄π΅ = 10, π΅πΆ = (π₯ + 1), πΆπ· = 20, πΈπΉ = 10, and πΉπΊ = 10. Find the value of π₯ and the length of πΊπ».

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### Video Transcript

In the diagram below, π΄π΅ equals 10, π΅πΆ equals π₯ plus one, πΆπ· equals 20, πΈπΉ equals 10, and πΉπΊ equals 10. Find the value of π₯ and the length of line segment πΊπ».

Letβs begin this question by filling on the given length measurements. We arenβt given any units of measurement, but these would all be length units. We are asked firstly to find the value of π₯ which appears as part of the length of line segment π΅πΆ. In order to work out the value of π₯ and indeed the length of the line segment πΊπ», weβll need to use these four parallel lines. Lines π΄πΈ, π΅πΉ, πΆπΊ, and π·π» weβre all given are parallel. We also note that the other two lines π΄π· and πΈπ» are both transversals. A transversal is a line that intersects two or more lines in the same plane at distinct points.

Knowing that we have these four parallel lines and two transversals means that we could apply Thaleβs theorem, or the basic proportionality theorem. This theorem tells us that if three or more parallel lines intersect two transversals, then they cut off the transversals proportionally. And so we can say that the ratio of the line segment π΄π΅ to that of π΅πΆ must be equal to the ratio of the length of the line segment πΈπΉ to that of the line segment πΉπΊ. And so we could write the proportion that π΄π΅ over π΅πΆ is equal to πΈπΉ over πΉπΊ.

We can then substitute in the given length measurements and see if we can solve it for π₯. This gives us 10 over π₯ plus one is equal to 10 over 10. We can then simplify the right-hand side, since 10 over 10 is simply equal to one. Next, we can multiply both sides by π₯ plus one. So we have 10 is equal to one times π₯ plus one. And, of course, one times π₯ plus one is just π₯ plus one. Subtracting one from both sides, we have nine is equal to π₯. And so weβve answered the first part of the question. Weβve worked out that π₯ is equal to nine.

For the second part of the question, we need to work out the length of the line segment πΊπ», which is here at the top part of the diagram. To work this out, we will apply the basic proportionality theorem once more. Now we know that the line segment of πΆπ· will be proportional to this line segment of πΊπ». We can write a proportionality statement involving these two line segments and either π΅πΆ and πΉπΊ or π΄π΅ and πΈπΉ. Letβs use the line segments π΄π΅ and πΈπΉ. We could therefore write the proportionality statement that π΄π΅ over πΆπ· is equal to πΈπΉ over πΊπ». When we substitute in the given lengths, we have 10 over 20 is equal to 10 over πΊπ».

We could then cross multiply and work out the length of πΊπ». However, we might also notice that the numerators on both sides of the equation are the same. Theyβre both 10. And so in order to be equal, then the denominators must be the same. We could therefore immediately work out that πΊπ» must be equal to 20.

Itβs worth pointing out that even though in this question we have a number of pairs of congruent line segments, but this wonβt always be the case. The proportionality theorem just tells us that these line segments have to be proportional. In this question, they were congruent as well. But now we can give the two answers: π₯ is equal to nine and πΊπ» is equal to 20 length units.

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