Given right triangle A BC with altitude B D drawn to hypotenuse AC. If AB = 6 and
AD = 4, what is the length of AC? And what is BD
In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments that are proportional to the two legs. In other words, if AD is the altitude and AB and BC are the legs, we have the following proportion:
AD/DB = AB/BC
Plugging in the given values, we have:
4/DB = 6/AC
Cross-multiplying, we get:
6DB = 4AC
Dividing both sides by 2, we have:
3DB = 2AC
Now, we need to find the length of AC. Since we have one equation and two unknowns (AC and DB), we cannot find the exact values. However, we can find their relationship.
To find the length of AC, we need to find the value of DB. We can do this by using another similar triangle created by the altitude.
Triangle ABD is similar to triangle BCD because they share angle B and angle A is a right angle.
By the similarity of triangles, we have the following proportion:
BD/AD = AC/BC
Plugging in the given values, we have:
BD/4 = AC/BC
Cross-multiplying, we get:
BCBD = 4AC
Dividing both sides by BC, we have:
BD = 4AC/BC
Now, we can substitute this value of BD back into our original equation to find the relationship between AC and DB.
3DB = 2AC
3(4AC/BC) = 2AC
12AC/BC = 2AC
Multiplying both sides by BC, we have:
12AC = 2AC(BC)
Dividing both sides by 2AC, we have:
BC = 6
Now, we can substitute this value of BC back into our equation for DB to find its value.
BD = 4AC/BC
BD = 4AC/6
BD = 2AC/3
Therefore, the length of AC is 6 units and the length of BD is 2AC/3 units.