We are also asked to formalize the conjecture that the product of the lengths of the two legs is equal to the product of the length of the hypotenuse and the length of the altitude to the hypotenuse.
2
&Given:&& â–ł ABC is a right triangle with
& && altitude to the hypotenuseCF
&Prove:&& AC* CB = AB* CF
b To prove the conjecture of Part A, let's recall that accroding to Theorem 7-3, the altitude to the hypotenuse divides a right triangle into two triangles that are similar to the original triangle. Let's use one of these similarities.
â–ł ABC~ â–ł CBF
Notice that â–ł CBF has sides appearing in the conjecture from Part A.
The hypotenuse CB of â–ł CBF is a leg of â–ł ABC.
The leg CF of â–ł CBF is the altitude to the hypotenuse of â–ł ABC.
Let's identify the sides in â–ł ABC that correspond to these sides of â–ł CBF.
Side in â–ł CBF
Corresponding side in â–ł ABC
CB
AB
CF
AC
We know that corresponding sides of similar triangles are proportional.
AC:CF=AB:CB
According to the Cross Products Property, the product of the extremes is equal to the product of the means in this proportion.
AC* CB = AB* CF
This completes the proof, so the conjecture is true. We can summarize the steps above in a flow proof.
Completed Proof
2
&Given:&& â–ł ABC is a right triangle with
& && altitude to the hypotenuseCF
&Prove:&& AC* CB = AB* CF
Proof:
