quadrilateral ABCD has right angles at B and D. If ABCD is kite-shaped with AB=AD=20 and BC=CD=15, find the radius of the circle inscribed in ABCD.
Join BD
BD will be a line of symmetry and the centre of the inscribed circle will have to be on AC and have to be the midpoint of BD. Let that point be P
BD^2 = 20^2 + 20^2 - 2(20)(20)cosA
= 800 - 800cosA
also BD^2 = 15^2+15^2-2(15)(15)cosC
= 450-450cosC
so 450-450cosC= 800-800cosA
800cosA -450cosC = 350
but C = 180-A
cosC = cos(180-A) = -cosA
800cosA +450cosA = 350
CosA = .28 (nice)
A = 73.7398°
BD^2 = 800-800cosA = 800-800(.28) = 576
BD = √576 = 24
BP = 12
Let r be the radius of the circle from P to line AB
angle ABD = 53.13..°
and BP, the hypotenuse of that little triangle is
sin 53.13...° = r/12
r = 12sin53.13..° = 12(.8) = 9.6
To find the radius of the circle inscribed in quadrilateral ABCD, we can make use of the fact that the perpendicular bisectors of the sides of a kite intersect at the center of the inscribed circle.
Here's how you can find the radius step-by-step:
Step 1: Draw the diagonals AC and BD.
Step 2: Observe that since angles B and D are right angles, the diagonals AC and BD are perpendicular bisectors of each other.
Step 3: The intersection point of the diagonals AC and BD is the center of the inscribed circle. Let's call it O.
Step 4: Since AB = AD = 20, the length of diagonal BD is twice the side length of the kite.
Step 5: The length of BD is equal to the length of AC, which is equal to 15 + 15 = 30.
Step 6: Therefore, the radius of the inscribed circle is equal to half the length of BD or AC.
Step 7: Since BD = 30, the radius of the inscribed circle is 30/2 = 15.
So, the radius of the circle inscribed in ABCD is 15 units.
To find the radius of the circle inscribed in quadrilateral ABCD, we need to use the property that the circle's center is the point of intersection of the diagonals.
Since ABCD is a kite, the diagonals AC and BD intersect at right angles at point O. Let's find the length of AC first.
Since AB = AD = 20, and BC = CD = 15, we can split ABCD into two congruent right triangles, namely ABD and CBD.
Using the Pythagorean theorem, we can find the length of AC:
AC^2 = AB^2 + BC^2
AC^2 = 20^2 + 15^2
AC^2 = 400 + 225
AC^2 = 625
AC = √625
AC = 25
Now that we know the length of AC, we can find the radius of the inscribed circle. The radius is given by the formula:
Radius = (Area of Quadrilateral ABCD) / (Semi-perimeter of Quadrilateral ABCD)
To find the area of ABCD, we can split it into two triangles, ABD and CBD. The area of each triangle can be calculated using the formula:
Area of Triangle = 0.5 * (length of base) * (length of perpendicular height)
Since ABD is a right triangle, the height is equal to BC = 15. The base is half of AC = 25/2 = 12.5. So the area of ABD is:
Area of ABD = 0.5 * 12.5 * 15
Area of ABD = 93.75
Since ABCD is a kite, the area of CBD is equal to the area of ABD, so the area of ABCD is:
Area of ABCD = 2 * Area of ABD
Area of ABCD = 2 * 93.75
Area of ABCD = 187.5
Next, we need to find the semi-perimeter of ABCD. It can be calculated by summing the lengths of the four sides and dividing by 2:
Semi-perimeter of ABCD = (AB + BC + CD + DA) / 2
Semi-perimeter of ABCD = (20 + 15 + 15 + 20) / 2
Semi-perimeter of ABCD = 70 / 2
Semi-perimeter of ABCD = 35
Now, we can calculate the radius of the inscribed circle:
Radius = (Area of ABCD) / (Semi-perimeter of ABCD)
Radius = 187.5 / 35
Radius ≈ 5.357
Thus, the radius of the circle inscribed in quadrilateral ABCD is approximately 5.357.