Video Transcript
Given that π΄π· equals π₯ centimeters, π·π΅ equals 30 centimeters, π΅πΈ equals π₯ plus seven centimeters, and πΈπΆ equals 18 centimeters, find the value of π₯.
In our figure, the larger triangle π΄π΅πΆ is being cut by the transversal π·πΈ. And weβre told that this line segment π·πΈ is parallel to the line segment π΄πΆ. And this should remind us of the side splitter theorem for triangles, which tells us if the line is parallel to a side of a triangle and the line intersects the other two sides, then the line divides those sides proportionately. So we can say the segment π΄π· is in proportion to the segment πΈπΆ and the segment π΅π· is in proportion to the segment π΅πΈ.
We can write this proportionality in the fraction form π΄π· over πΈπΆ is equal to π·π΅ over π΅πΈ. And then we can label our diagram with the information we were given. π΄π· equals π₯ centimeters, π·π΅ equals 30 centimeters, π΅πΈ equals π₯ plus seven centimeters, πΈπΆ equals 18 centimeters. And then we can plug these values into the proportional statement we set up. π₯ over 18 is then equal to 30 over π₯ plus seven. From there, we cross multiply to see if we can solve for π₯. π₯ times π₯ plus seven equals π₯ squared plus seven π₯. And 18 times 30 equals 540. Since we end up with a quadratic equation, we wanna set that equal to zero and see if we can solve by factoring. So we subtract 540 from both sides of the equation. And we get π₯ squared plus seven π₯ minus 540 equals zero.
We want to try to break these up into two terms that multiply together to equal negative 540 and sum together to positive seven. Letβs first consider some factors of 540. Almost immediately, we recognize that 540 is divisible by 10. 54 times 10 is 540. However, we need these two factors to sum to positive seven, which means weβre looking for factors that have an absolute value that are closer to each other. So because I know that 54 is even, I know that 20 will be a factor of 540. If we divide 540 by 20, we get 27. 20 times 27 equals 540, which means negative 20 times 27 equals negative 540. And when we add negative 20 and 27, we get positive seven.
These are the factors weβre looking for. We want negative 20 and positive 27. Our two factors are then π₯ minus 20 and π₯ plus 27. We set both of these values equal to zero. And we find π₯ equals 20 or π₯ equals negative 27. However, π₯ cannot equal negative 27 in this case as weβre dealing with distance. Since negative 27 is not a valid solution, the only thing π₯ can be here is 20. Itβs probably worth checking our proportions here. Since π₯ equals 20, π΄π· equals 20. We would have 20 over 18 is equal to 30 over 27. 20 over 18 reduces to 10 over nine. And if we divide 30 by three, we get 10. 27 divided by three equals nine. Both of these proportions reduce to 10 over nine and confirm that our solution of π₯ equals 20 does make the lengths proportional.