Find the length of line 𝐵𝐷.
As we look at this image, we can
see three distinct triangles: the large triangle on the outside, a small right
triangle inside, and a second larger right triangle in the inside. Our first step will be to find the
length of line 𝐵𝐶. 𝐵𝐶 is one of the sides of the
largest triangle. We can use the Pythagorean theorem
to find the length of 𝐵𝐶.
The hypotenuse the largest side of
this triangle is five. Five is our 𝑐 value. Five squared is equal to four
centimeters squared, four squared, plus 𝑏 squared, which is our unknown value, the
length of side 𝐵𝐶. Five squared equals 25. Four squared equals 16. Bring down the 𝑏 squared.
To get 𝑏 squared by itself, I’ll
subtract 16 from both sides of the equation. 25 minus 16 equals nine. 𝑏 squared equals nine. We can take the square root of both
sides of the equation. The square root of nine is three
the square root of 𝑏 squared is 𝑏. This tells us that the side length
of line 𝐵𝐶 equals three centimeters. The main length we’re interested in
finding is line 𝐵𝐷, and that will be a little bit more complicated.
I’m just gonna move all of this
information over to make a little room. The next thing we wanna do is break
up these two triangles: triangle one and triangle two. The next thing I wanna do is label
all the sides. We’re gonna call the length of 𝐵𝐷
letter 𝑎, and we’re gonna call the length of 𝐶𝐷 letter 𝑏. So for triangle one, 𝑎 squared
plus 𝑏 squared equals three squared.
We’re breaking up this
five-centimeter length into two pieces. One of the pieces measures 𝑏,
we’ve said. And if triangle one’s portion
measures 𝑏, then triangle two’s side would measure five centimeters minus 𝑏. So follow me here, the side lengths
of triangle two are 𝑎 squared plus five minus 𝑏 squared equals four squared.
We can go ahead and factor five
minus 𝑏 squared. Five minus 𝑏 times five minus
𝑏. We can FOIL: five times five equals
25, five times negative 𝑏 equals negative five 𝑏, negative 𝑏 times five equals
negative five 𝑏, negative 𝑏 times negative 𝑏 equals positive 𝑏 squared. By combining like terms, we find
that five minus 𝑏 squared equals 25 minus 10𝑏 plus 𝑏 squared.
We add that to our equation. Now we have two equations that will
help us solve for 𝑎. We’re going to solve this by using
substitution. In the formula for triangle one,
I’m going to subtract 𝑏 squared from both sides of the equation. On the left, we have 𝑎 squared,
the 𝑏s cancel out, 𝑎 squared is equal to three squared minus 𝑏 squared. Now we’ll go ahead and square our
three. Three squared equals nine.
𝑎 squared equals nine minus 𝑏
squared. We’re going to take this value for
𝑎 squared and plug it in to the equation from triangle two. Instead of having 𝑎 squared in the
formula, we’ll now use nine minus 𝑏 squared. And then the rest of the equation,
we’ll copy down exactly as it’s written. We can drop the parentheses.
Our next step will be to combine
like terms: nine plus 25 equals 34, negative 𝑏 squared plus positive 𝑏 squared
equal zero. They cancel each other out. There’s nothing to combine negative
10𝑏 with, so we bring it down. And then we copy down what’s to the
right of the equal sign: four squared. Four squared equals 16. We need to get 𝑏 by itself, so we
subtract 34 from both sides of the equation.
Positive 34 minus 34 equals
zero. 16 minus 34 equals negative 18. We bring down what’s on the left
side. And now we have an equation that
says negative 10𝑏 equals negative 18. We divide both sides by negative
10. Negative 10 divided by negative 10
equals one. So we’re left with 𝑏 equals on the
left, negative 18 divide by negative 10 equals positive 1.8.
Since 𝑏 equals positive 1.8, line
segment 𝐶𝐷 is equal to 1.8 centimeters and line segment 𝐷𝐴 is equal to five
centimeters minus 1.8 centimeters, 3.2 centimeters. But the value we really need to
know is the value of 𝑎. We know that 𝑎 squared is equal to
nine minus 𝑏 squared, and we know that 𝑏 equals 1.8.
That means 𝑎 squared is equal to
nine minus 1.8 squared. 1.8 squared is 3.24. Nine minus 3.24 equals 5.76. 𝑎 squared equals 5.76. We take the square root of both
sides. The square root of 5.76 equals
2.4. And that means 𝑎 equals 2.4
centimeters. The measure of line segment 𝐵𝐷
equals 2.4 centimeters.