The circle with center O is inscribed in kite ABCD. AB is 24 cm and BC is 18 cm. D is 90 degrees.

Find the length of the radius if the circle, in cm.

Guest Jan 10, 2018

#2**+1 **

Ahh! Finally!!!! I think I found another way!!!!

Here's my own drawing...it's not as accurate as CPhill's :

Each of the purple lines is a radius of the circle, " r ". Each radius meets the sides of the kite at right angles because each side of the kite is tangent to the circle.

Notice that there are two kites formed within the big kite. We can be sure that both these inside kites are similar to kite ABCD because all the corresponding angles are the same.

Looking at the kite ABCD to the little kite, we can say...

\(\frac{24}{18}\,=\,\frac{r}{18-r} \\~\\ 24(18-r)\,=\,18r \\~\\ 432-24r\,=\,18r \\~\\ 432\,=\,42r \\~\\ \frac{432}{42}\,=\,r \\~\\ r\,=\,\frac{72}{7}\qquad\text{cm}\) Cross multiply

hectictar
Jan 11, 2018

#1**+2 **

Here's one method [ but maybe not the fastest or easiest ]

Let AC = √ [ 24^2 + 18^2] = √ 900 = 30

So....let A = (-15, 0) and C = (15, 0)

And we can find B and D by intersecting the circles

(x + 15)^2 + y^2 = 24^2 and

(x - 15)^2 + y^2 = 18^2

So....the intersection of these (arbitrarily) gives B = (4.2, 14.4) and D = (4.2, -14.4)

Here's a pic :

Let the center of the circle lie at ( E, 0) [ I'm using E for the center instead of O ]

The slope of AB is [ 14.4] / [ 4.2 - - 15] = 14.4 / 19.2 = 3/4

And the equation of this line segment is

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

y = (3/4)x + 45/4

4y = 3x + 45

3x - 4y + 45 = 0

And the slope of BC is -4/3

And the equation of this line segment is

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

y = (-4/3)x + 20

3y = -4x + 60

4x + 3y - 60 = 0

And the distance between a point (m, n) and the line Ax + By + C = 0 is given by

l Am + Bn + C l / √ [ A^2 + B^2 ]

So we want to equate these and solve for E

l 3E - 4 (0) + 45 l / √ [ 3^2 + (-4)^2] = l 4E + 3(0) - 60 l / √[4^2 + 3^2 ]

l 3E + 45 l / 5 = l 4E - 60 l / 5

We have these two possible equations

3E + 45 = 4E - 60

E = 105 reject

Or

3E + 45 = - [ 4E - 60 ]

3E + 45 = -4E + 60

7E = 15

E = 15/7

So....the center of of the circle is (E, 0) = (15/7, 0)

And using l 3C + 45 l / 5....the radius of the circle is

l 3 (15/7) + 45 l / 5 = l 45/7 + 45 l / 5 = 45/35 + 9 = 9/7 + 9 = 72/7 cm

So.....the equation of the inscribed circle is

(x - 15/7)^2 + y^2 = (72/7)^2

CPhill
Jan 10, 2018

#2**+1 **

Best Answer

Ahh! Finally!!!! I think I found another way!!!!

Here's my own drawing...it's not as accurate as CPhill's :

Each of the purple lines is a radius of the circle, " r ". Each radius meets the sides of the kite at right angles because each side of the kite is tangent to the circle.

Notice that there are two kites formed within the big kite. We can be sure that both these inside kites are similar to kite ABCD because all the corresponding angles are the same.

Looking at the kite ABCD to the little kite, we can say...

\(\frac{24}{18}\,=\,\frac{r}{18-r} \\~\\ 24(18-r)\,=\,18r \\~\\ 432-24r\,=\,18r \\~\\ 432\,=\,42r \\~\\ \frac{432}{42}\,=\,r \\~\\ r\,=\,\frac{72}{7}\qquad\text{cm}\) Cross multiply

hectictar
Jan 11, 2018