geometry problem......?

Question

DEFG is a rectangle with width of 10 units and lenght of 24 units. A diagonal is drawn. A circle is inscribed in each triangle. Find the distance between centers A and B. Express answer in radical form. ( I am now on my 1st hour but I can't get the answer)

Answer 1

I think it's 2*sqrt(65), but a bit hard to explain without a diagram. No doubt you've already drawn a few.

Look at one of the triangles. The diagonal and two sides of the rectangle are tangents to the inscribed circle, whose points of contact divide each of these sides into two unequal bits. Call the radius of the circle r. Then the two "bits" from the corner of the rectangle are both r, and the other bits of these sides are 10-r and 24-r. The diagonal is 26 (Pythag) and call the shorter of its bits b, then the longer is 26-b.

Now since the tangents to a circle from an external point are equal, we have
10 - r = b
24 - r = 26 - b

Subtracting these gives
b = 6, and so r = 4.

The part of the diagonal between the points of contacts of the two circles now becomes 26 - 2*6, i.e. 14.

From one circle, draw a radius to the diagonal (4 units long) and continue it another 4 units, then join to the centre of the other circle, thus completing two sides of a rightangled triangle with sides 8 and 14, and hypotenuse the distance between the circles.

Thus this distance is
sqrt(8^2 + 14^2)
=sqrt(260)
=sqrt(4*65)
=2*sqrt(65)

Answer 2

Start by drawing the picture, with the 2 circles. Notice the distance between them is the hypotenuse of a right triangle with legs coming out of the circles' radii. The 2 legs of this triangle are of size 24 - 2r and 10 - 2r.

Set up a coordinate system through a vertex of the rectangle. Then the equation of the diagonal is y = -10/24x + 10, or 24y + 10x - 240 = 0.

The formula for the distance between a point (a, b) and a line Ax + By + C = 0 is (Aa + Bb + C)/sqrt(A^2 + B^2). So look at the point (r, r) at the center of one circle. The distance from the line is r, so

(24r + 10r - 240)/(24^2 + 10^2) = +-r
24r + 10r - 240 = +-26r

so 34r +- 26r = 240

8r = 240 or 60r = 240
r = 30 or 4. Clearly 30 is wrong (it is on the other side), so 4 must be the answer.

If you look at the small triangle in the middle, its hypotenuse is the distance you want, and the legs are 10 - 2r and 24 - 2r, so if r = 4 then the legs are 2 and 16, the distance is sqrt(260) = 2sqrt(65)

Answer 3

The following formula relates the radius of the inscribed circle to the perimeter and area of the triangle.

area of triangle = 0.5 radius perimeter of triangle

0.5*10*24 = 0.5 r (10+24 + sqrt(10*10+24*24))
where the sqrt is for the length of the hypotenuse using Pythagorean theorem.
Thus,
120=0.5 r (34+26) or
r = 4

Since the center has the same distance from both legs, the distance between both centers is sqrt([24-(2*4)]^2+[10-(2*4)]^2)
= sqrt(16^2 + 2^2) = sqrt(4*65) = 2sqrt(65).

Answer 4

Assign coordinates D(0,0), E(24,0), F(24,10), and G(0,10)
Draw a diagonal GE from (0,10) to (24,0).
Observe that tan(EGF/2) = tan(DEG/2) = 0.2
We now can define 4 lines that will intersect at the centers of our 2 circles:
y = x
-0.2 = (y - 0)/(x - 24)
-0.2x + 4.8 = y
-y = 0.2x - 4.8
1.2x = 4.8
x = 4
y = 4

1 = (y - 10)/(x - 24)
-0.2 = (y - 10)(x - 0)
x = 20
y = 6
d = √((20-4)^2 + (6 - 4)^2)
d = √(16^2 + 2^2)
d = 2√(8^2 + 1^2)
d = 2√65

The circles, or course, have a radius of 4.

Answer 5

I undergo in innovations that there is a thorem that if the two aspects of a traingle are equivalent then the attitude opposite to them additionally are same. Please see the thorem. So the bisectors make a isoscele traingle with the component in between the angles have been getting bisected.i will later write you the evidence of the theorm. because of the fact the each and each a million/2 of the angles are equivalent, then the entire angles are equivalent. right here you chanced on the two angles are equivalent. As in line with definition: in a traingle if the two angles are equivalent, then that traingle is reported as a isoscele traingle.

Answer 6

I think the answer is 2square roots of 15.