WEBVTT
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In the figure, π΄π· equals 10, π΄πΈ equals five, π΅π· equals four π₯ plus two, and πΈπΆ equals π₯ plus three.
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What are the lengths of the line segment π΄π΅ and the line segment πΈπΆ?
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Well, the first thing we want to do is mark on all the information weβve been given.
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So weβve got π΄π· equals 10, π΄πΈ equals five, π΅π· equals four π₯ plus two, and πΈπΆ equals π₯ plus three.
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Now, if we take a look at the shape itself, then what we actually have are two triangles, triangle π΄π·πΈ and triangle π΄π΅πΆ.
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And we also have two parallel lines.
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But we can use something about these parallel lines to tell us something about our triangles.
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Well, as our parallel lines are transversed by the line π΄πΆ, we can say that the angles at πΆ and πΈ are corresponding.
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Similarly, if we look at the line π΄π΅, which transverses our two parallel lines, then this means that weβre gonna have corresponding angles at π· and π΅, so these are gonna be the same.
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And then finally, both triangles have a shared angle at π΄.
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Okay, so what does this mean?
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Well, what it tells us is that using the proof AAA or angle-angle-angle, then triangle π΄π·πΈ is similar to triangle π΄π΅πΆ.
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Well, what do we know about similar triangles?
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Well, actually, what it means is that they are in fact an enlargement of one another.
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So therefore, corresponding sides are always proportional.
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So therefore, if we take a look at our triangle, we can say that π΄π΅ over π΄π· is gonna be equal to π΄πΆ over π΄πΈ.
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And as we said, itβs because our corresponding sides are proportional.
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So therefore, what weβre gonna have is π΄π΅, which is gonna be four π₯ plus two plus 10 because itβs π·π΅ and π΄π·, and then over 10, because thatβs our π΄π·.
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And itβs gonna be equal to π΄πΆ, which is π₯ plus three plus five, over π΄πΈ, which is five.
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Okay, great.
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So now what we could do is solve this equation to find out what π₯ is.
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So next, what weβre gonna do is multiply through both sides by 10.
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And thatβs because we want to remove the fractional element of our equation.
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And when we do that, weβre gonna get four π₯ plus 12 equals two multiplied by π₯ plus eight.
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And then if you divide three by two, you get two π₯ plus six equals π₯ plus eight.
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So then, if we subtract π₯ and subtract six from each side of the equation, weβre gonna get π₯ equals two.
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Okay, great.
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So weβve now found our value for π₯.
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Well, if we take a look at π₯, what we can see is that we want to find the lengths of the line segment π΄π΅ and the line segment πΈπΆ.
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Well, weβve already identified that π΄π΅ is equal to four π₯ plus two plus 10, which is four π₯ plus 12.
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Well, as weβve identified that π₯ is equal to two, weβre gonna substitute this in.
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So weβre gonna get four multiplied by two plus 12.
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So weβve answered the first part of the question because π΄π΅ is equal to 20.
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Okay, so now letβs move on to πΈπΆ.
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Well, we can see that πΈπΆ is π₯ plus three.
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So if we substitute in π₯ equals two, itβs gonna be two plus three, which is gonna give us five.
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So therefore, we can say that the line segments π΄π΅ and πΈπΆ have lengths of 20 and five, respectively.