Video: Finding the Length of the Altitude in a Right-Angled Triangle given the Triangle’s Dimensions

Find the length of line 𝐡𝐷.


Video Transcript

Find the length of line 𝐡𝐷.

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 𝐡𝐢. 𝐡𝐢 is one of the sides of the largest triangle. We can use the Pythagorean theorem to find the length of 𝐡𝐢.

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 𝐡𝐢. 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 𝐡𝐢 equals three centimeters. The main length we’re interested in finding is line 𝐡𝐷, 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 𝐡𝐷 letter π‘Ž, and we’re gonna call the length of 𝐢𝐷 letter 𝑏. So for triangle one, π‘Ž 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 π‘Ž 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 π‘Ž. 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 π‘Ž squared, the 𝑏s cancel out, π‘Ž squared is equal to three squared minus 𝑏 squared. Now we’ll go ahead and square our three. Three squared equals nine.

π‘Ž squared equals nine minus 𝑏 squared. We’re going to take this value for π‘Ž squared and plug it in to the equation from triangle two. Instead of having π‘Ž 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 𝐢𝐷 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 π‘Ž. We know that π‘Ž squared is equal to nine minus 𝑏 squared, and we know that 𝑏 equals 1.8.

That means π‘Ž squared is equal to nine minus 1.8 squared. 1.8 squared is 3.24. Nine minus 3.24 equals 5.76. π‘Ž 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‘Ž equals 2.4 centimeters. The measure of line segment 𝐡𝐷 equals 2.4 centimeters.

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