Challenge problems: circumscribing shapes

Solve two challenging problems that apply properties of tangents to find the perimeter of a circumscribing shape.

Problem 1

All sides of triangle, A, B, C are tangent to circle P.
What is the perimeter of triangle A, B, C?
  • Dit svar bør være
  • et heltal, som 6
  • en forkortet, ægte brøk, som eksempelvis 3, slash, 5
  • en forkortet, uægte brøk, som eksempelvis 7, slash, 4
  • et blandet tal, som eksempelvis 1, space, 3, slash, 4
  • et præcist decimaltal, som eksempelvis 0, point, 75
  • et multiplum af pi, som f.eks. 12, space, p, i eller 2, slash, 3, space, p, i
units

Any two segments tangent to a circle from a common endpoint are congruent.
With this information, we can identify another segment which is also 16 units long.
Let's use the variable x to represent the distance from point C to the circle. By the same logic as above, we can label the rest of the lengths in the perimeter.
Now we can find the perimeter by adding up the lengths of the sides:
empty space, start color purple, 14, minus, x, end color purple, plus, start color green, x, end color green, plus, start color green, x, end color green, plus, start color pink, 16, end color pink, plus, start color pink, 16, end color pink, plus, start color purple, 14, minus, x, end color purple
equals, 60
The perimeter of the triangle is 60 units long.

Problem 2

All sides of quadrilateral A, B, C, D are tangent to circle P.
What is the perimeter of quadrilateral A, B, C, D?
  • Dit svar bør være
  • et heltal, som 6
  • en forkortet, ægte brøk, som eksempelvis 3, slash, 5
  • en forkortet, uægte brøk, som eksempelvis 7, slash, 4
  • et blandet tal, som eksempelvis 1, space, 3, slash, 4
  • et præcist decimaltal, som eksempelvis 0, point, 75
  • et multiplum af pi, som f.eks. 12, space, p, i eller 2, slash, 3, space, p, i
units

Any two segments tangent to a circle from a common endpoint are congruent.
With this information, we can identify two other segments which are 3, point, 7 and 9, point, 6 units long, respectively.
Let's use the variable x to represent the distance from point C to the circle. By the same logic as above, we can label the rest of the lengths in the perimeter.
Now we can find the perimeter by adding up the lengths of the sides:
empty space, start color blue, 12, minus, x, end color blue, plus, start color blue, 12, minus, x, end color blue, plus, start color purple, x, end color purple, plus, start color purple, x, end color purple, plus, start color pink, 9, point, 6, end color pink, plus, start color pink, 9, point, 6, end color pink, plus, start color green, 3, point, 7, end color green, plus, start color green, 3, point, 7, end color green
equals, 50, comma, 6
The perimeter of the quadrilateral is 50, point, 6 units long.