Video Transcript
The perimeter of a triangle is
negative four 𝑥 to the fourth power 𝑦 to the fifth power plus nine 𝑥 squared 𝑦
minus three 𝑥 to the fifth power 𝑦 squared centimeters. And the lengths of two of its sides
are negative five 𝑥 to the fourth power 𝑦 to the fifth power minus six 𝑥 squared
𝑦 centimeters and two 𝑥 squared 𝑦 plus four 𝑥 to the fifth power 𝑦 squared
centimeters. Express the length of the third
side in terms of 𝑥 and 𝑦.
We begin by recalling that the
perimeter of a triangle can be calculated by finding the sum of its three side
lengths. In this question, we are given
expressions for the lengths of two of the sides of the triangle in terms of the
variables 𝑥 and 𝑦. And we are also given an expression
for the perimeter of the triangle in terms of these two variables. We are asked to find an expression
for the length of a third side, which we will call 𝑠.
Using this information, we can
write the following equation, where the sum of the three side lengths is equal to
the perimeter. Next, we can remove the parentheses
and collect the like terms on the left-hand side. Recalling that like terms must have
the same variables raised to the same powers, we can see that there is one such pair
on the left-hand side of our equation. Negative six 𝑥 squared 𝑦 plus two
𝑥 squared 𝑦 is equal to negative four 𝑥 squared 𝑦. And as such the left-hand side
simplifies to 𝑠 minus five 𝑥 to the fourth power 𝑦 to the fifth power minus four
𝑥 squared 𝑦 plus four 𝑥 to the fifth power 𝑦 squared.
Next, to isolate 𝑠, we need to add
five 𝑥 to the fourth power 𝑦 to the fifth power and four 𝑥 squared 𝑦 to both
sides, as well as subtracting four 𝑥 to the fifth power 𝑦 squared from both
sides. This gives us 𝑠 is equal to
negative four 𝑥 to the fourth power 𝑦 to the fifth power plus nine 𝑥 squared 𝑦
minus three 𝑥 to the fifth power 𝑦 squared plus five 𝑥 to the fourth power 𝑦 to
the fifth power plus four 𝑥 squared 𝑦 minus four 𝑥 to the fifth power 𝑦
squared.
We are now in a position to collect
like terms on the right-hand side. We recall that like terms have the
same variables raised to the same powers. There are three such pairs in this
case, underlined in orange, pink, and green. Firstly, we have two terms
containing 𝑥 to the fourth power and 𝑦 to the fifth power. Negative four 𝑥 to the fourth
power 𝑦 to the fifth power plus five 𝑥 to the fourth power 𝑦 to the fifth power
is equal to 𝑥 to the fourth power 𝑦 to the fifth power. Secondly, there are two terms
containing 𝑥 squared 𝑦. Nine 𝑥 squared 𝑦 plus four 𝑥
squared 𝑦 is equal to 13𝑥 squared 𝑦.
Finally, there are two terms
containing 𝑥 to the fifth power 𝑦 squared. And negative three 𝑥 to the fifth
power 𝑦 squared minus four 𝑥 to the fifth power 𝑦 squared is equal to negative
seven 𝑥 to the fifth power 𝑦 squared. So 𝑠 is equal to 𝑥 to the fourth
power 𝑦 to the fifth power plus 13𝑥 squared 𝑦 minus seven 𝑥 to the fifth power
𝑦 squared.
We can therefore conclude that the
length of the third side of the triangle in terms of 𝑥 and 𝑦 is 𝑥 to the fourth
power 𝑦 to the fifth power plus 13𝑥 squared 𝑦 minus seven 𝑥 to the fifth power
𝑦 squared centimeters.
