Video Transcript
Given that 𝐴𝐷 equals 𝑥 centimeters, 𝐷𝐵 equals 30 centimeters, 𝐵𝐸 equals 𝑥 plus seven centimeters, and 𝐸𝐶 equals 18 centimeters, find the value of 𝑥.
In our figure, the larger triangle 𝐴𝐵𝐶 is being cut by the transversal 𝐷𝐸. And we’re told that this line segment 𝐷𝐸 is parallel to the line segment 𝐴𝐶. 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 𝐴𝐷 is in proportion to the segment 𝐸𝐶 and the segment 𝐵𝐷 is in proportion to the segment 𝐵𝐸.
We can write this proportionality in the fraction form 𝐴𝐷 over 𝐸𝐶 is equal to 𝐷𝐵 over 𝐵𝐸. And then we can label our diagram with the information we were given. 𝐴𝐷 equals 𝑥 centimeters, 𝐷𝐵 equals 30 centimeters, 𝐵𝐸 equals 𝑥 plus seven centimeters, 𝐸𝐶 equals 18 centimeters. And then we can plug these values into the proportional statement we set up. 𝑥 over 18 is then equal to 30 over 𝑥 plus seven. From there, we cross multiply to see if we can solve for 𝑥. 𝑥 times 𝑥 plus seven equals 𝑥 squared plus seven 𝑥. 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 𝑥 squared plus seven 𝑥 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. Let’s 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 we’re 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 we’re looking for. We want negative 20 and positive 27. Our two factors are then 𝑥 minus 20 and 𝑥 plus 27. We set both of these values equal to zero. And we find 𝑥 equals 20 or 𝑥 equals negative 27. However, 𝑥 cannot equal negative 27 in this case as we’re dealing with distance. Since negative 27 is not a valid solution, the only thing 𝑥 can be here is 20. It’s probably worth checking our proportions here. Since 𝑥 equals 20, 𝐴𝐷 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 𝑥 equals 20 does make the lengths proportional.
