1. Let \(\triangle ABC\) be a right triangle such that \(B \) is a right angle. A circle with diameter of \(BC\) meets side \(AC\) at \(D\) If the area of \(\triangle ABC\) is \(150\)and\(AC = 25,\) then what is \(BD\)?

tertre Mar 31, 2018

#1**+2 **

This one isn't too bad, tertre !!!

Note that a multiple of a 3-4-5 right triangle is a 15-20-25 right triangle....

Let B = (0,0)

A= (0,15)

C= (20, 0)

The area of ABC is (1/2)BC * BA = (1/2)(20)(15) = 150 which is what we need!!!

Let the circle with the diameter of BC have the equation (x - 10)^2 + y^2 = 100 (1)

And let the slope of the line containing AD = -15/20 = -3/4

And the equation of this line is

y =( -3/4) x + 15 or

y = (15 - 3/4)x

Square both sides of this

y^2 = ( 15 - (3/4)x)^2 (1)

Our objective is to find the x coordinate of the second intersection of AC and this circle

Sub (2) into (1) and we have

(x - 10)^2 + (15 - (3/4)x)^2 = 100 simplify

x^2 -20x + 100 + 225 - (90/4) x + (9/16) x^2 = 100

x^2 -20x + 225 - (90/4) x + (9/16)x^2 = 0 multiply through by 16

16x^2 - 320x + 3600 - 360x + 9x^2 = 0

25x^2 - 680x + 3600 = 0

Believe it or not....we can factor this as

(5x - 36 (5x - 100) = 0

Set each factor to 0 ans solve for x and we have that

x =36/5 and x = 20

We already know that the second value is an x intersection of the line and the circle

We are interested in the first.... and this is the x coordinate of D

The y coordinate is

y = ( - 3/4) (36/ 5) + 15

y = -108/20 + 15

y = -27/ 5 + 75/ 5

y = 48/ 5

So.....D = ( 36/5, 48/5)

So BD is easy to find as

sqrt [ (36/5)^2 + (48/ 5)^2 ] =

sqrt [ 36^2 + 48^2] / 25 =

sqrt [ 3600 ] / 5 =

60 / 5 =

12 units

Note something interesting....that triangle BDC is a 12-16-20 right triangle which is another multiple of a 3-4-5 right triangle !!

Here's a pic of all of this :

CPhill Mar 31, 2018

#3**0 **

Since we know triangle ABC is a 15-20-25 right-angle triangle, then we can easily find the x coordinate of the second intersection of AC and this circle as follows:

(150*2) / 25^2 =12/25 x 15 =7.2

(150*2) / 25^2 =12/25 x 20 =9.6, then D =7.2 and 9.6 and:

BD^2 = 7.2^2 + 9.6^2

BD^2 =144

**BD =12**

Guest Apr 1, 2018