# Video: Finding the Length of the Altitude in a Right-Angled Triangle given the Triangleβs Dimensions

Find the length of line π΅π·.

06:46

### Video Transcript

Find the length of line π΅π·.

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 π΅πΆ. π΅πΆ is one of the sides of the largest triangle. We can use the Pythagorean theorem to find the length of π΅πΆ.

The hypotenuse the largest side of this triangle is five. Five is our π value. Five squared is equal to four centimeters squared, four squared, plus π squared, which is our unknown value, the length of side π΅πΆ. Five squared equals 25. Four squared equals 16. Bring down the π squared.

To get π squared by itself, Iβll subtract 16 from both sides of the equation. 25 minus 16 equals nine. π 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 π squared is π. This tells us that the side length of line π΅πΆ equals three centimeters. The main length weβre interested in finding is line π΅π·, and that will be a little bit more complicated.

Iβ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. Weβre gonna call the length of π΅π· letter π, and weβre gonna call the length of πΆπ· letter π. So for triangle one, π squared plus π squared equals three squared.

Weβre breaking up this five-centimeter length into two pieces. One of the pieces measures π, weβve said. And if triangle oneβs portion measures π, then triangle twoβs side would measure five centimeters minus π. So follow me here, the side lengths of triangle two are π squared plus five minus π squared equals four squared.

We can go ahead and factor five minus π squared. Five minus π times five minus π. We can FOIL: five times five equals 25, five times negative π equals negative five π, negative π times five equals negative five π, negative π times negative π equals positive π squared. By combining like terms, we find that five minus π squared equals 25 minus 10π plus π squared.

We add that to our equation. Now we have two equations that will help us solve for π. Weβre going to solve this by using substitution. In the formula for triangle one, Iβm going to subtract π squared from both sides of the equation. On the left, we have π squared, the πs cancel out, π squared is equal to three squared minus π squared. Now weβll go ahead and square our three. Three squared equals nine.

π squared equals nine minus π squared. Weβre going to take this value for π squared and plug it in to the equation from triangle two. Instead of having π squared in the formula, weβll now use nine minus π squared. And then the rest of the equation, weβll copy down exactly as itβs written. We can drop the parentheses.

Our next step will be to combine like terms: nine plus 25 equals 34, negative π squared plus positive π squared equal zero. They cancel each other out. Thereβs nothing to combine negative 10π with, so we bring it down. And then we copy down whatβs to the right of the equal sign: four squared. Four squared equals 16. We need to get π 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 whatβs on the left side. And now we have an equation that says negative 10π equals negative 18. We divide both sides by negative 10. Negative 10 divided by negative 10 equals one. So weβre left with π equals on the left, negative 18 divide by negative 10 equals positive 1.8.

Since π equals positive 1.8, line segment πΆπ· is equal to 1.8 centimeters and line segment π·π΄ is equal to five centimeters minus 1.8 centimeters, 3.2 centimeters. But the value we really need to know is the value of π. We know that π squared is equal to nine minus π squared, and we know that π equals 1.8.

That means π squared is equal to nine minus 1.8 squared. 1.8 squared is 3.24. Nine minus 3.24 equals 5.76. π squared equals 5.76. We take the square root of both sides. The square root of 5.76 equals 2.4. And that means π equals 2.4 centimeters. The measure of line segment π΅π· equals 2.4 centimeters.