Triangle ABC is inscribed in a circle such that A and B are the endpoints of a diameter. CD is the altitude drawn to AB. If AC=15 and BD=16, compute the value of (AD*BC/CD)-3
To find the value of (AD * BC / CD) - 3, we need to first find the values of AD, BC, and CD.
Since A and B are endpoints of a diameter, triangle ABC is a right triangle with C as the right angle.
We are given that AC = 15 and BD = 16, so we can use the Pythagorean theorem to find CD.
The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
In this case, CD is the hypotenuse, and AC and BD are the other two sides.
So we have:
AC^2 + BD^2 = CD^2
(15^2) + (16^2) = CD^2
225 + 256 = CD^2
481 = CD^2
Taking the square root of both sides, we find:
CD = √481
Now that we have the value of CD, we can use similar triangles to find the values of AD and BC.
Since CD is perpendicular to AB, triangle ACD and triangle BCD are similar to triangle ABC.
Using the property of similar triangles, we have the following ratios:
AD / AC = CD / BC
AD = (AC * CD) / BC
Similarly, BC = (BD * CD) / AC
Plugging in the given values, we have:
AD = (15 * √481) / BC
BC = (16 * √481) / 15
Now, let's substitute these values into (AD * BC / CD) - 3:
((15 * √481) / BC) * ((16 * √481) / 15) / √481 - 3
Simplifying further:
(15 * 16 * √481 * √481) / (BC * 15 * √481) - 3
(16 * 481) / BC - 3
Replacing BC with its value:
(16 * 481) / ((16 * √481) / 15) - 3
Canceling out some terms:
(481 * 15) / √481 - 3
Simplifying:
7225 / √481 - 3
Finally, evaluate this expression using a calculator to find the numerical value.
C is the right angle => triangles ADC and CDB are similar.
AD/AC=CD/BC =>(AD*BC/CD)=AC
if I follow your work then
AC is 15 -3
Final answer 12