Find the length of the altitude drawn and the length of the two segments in which the hypotenuse is divided. Notice that when you rotate the triangle around the hypotenuse you get two cones.
See solution.
Practice makes perfect
Let's begin by labeling the vertices of the given triangle.
Before performing the rotations, let's find CD, AD, and BD. The Segment Addition Postulate gives us that AD+DB=25. Let's write two more equations by applying the Pythagorean Theorem to â–ł ADC and â–ł BDC.
AD+DB = 25 & (I) AD^2+CD^2 = 15^2 & (II) DB^2+CD^2=20^2 & (III)
To solve the system above, we will subtract Equation (II) from Equation (III), so we cancel CD^2.
DB^2 + CD^2 &= 20^2
^-AD^2 + CD^2 &= 15^2
BD^2 - AD^2 &= 175
Let's continue simplifying the resulting equation.
Now we can add the latter equation with Equation (I) and get the value of BD.
AD + DB &= 25
^+ BD-AD &= 7
2BD &= 32
From the above we have that BD=16, which implies that AD = 9. Finally, let's substitute AD=9 into Equation (II) to find the value of CD.
In this third case, let's rotate â–ł ABC around AB.
This time, we got a composite solid that is formed by two cones. Thus, the volume of this solid is the sum of the volumes of the two cones.
V = V_(left cone) + V_(right cone)
The left-hand side cone has a base radius of r=CD=12 and height h=AD=9. With this data we can find its volume.
