All radii in one circle are equal
The figure on the right shows a square drawn inside a circle. The perimeter of square ㄱㄴㄷㄹ is . What is the perimeter of triangle ㄹㄴㄷ, in cm?
Show solution
Understand
A square is drawn inside a circle so that all four corners sit on the circle. The square's perimeter is 40 cm, and the radius from the circle's center to a corner is 7 cm. The diagonal of the square passes through the center, so it is a diameter. I need the perimeter of the triangle formed by one side of the square, an adjacent side, and that diagonal.
- Square ㄱㄴㄷㄹ (ABCD) is inscribed in the circle, all four vertices on the circle.
- Perimeter of the square is 40 cm.
- Diagonal ㄹㄴ passes through the center and is a diameter of the circle.
- The radius from the center to a vertex is 7 cm.
- The perimeter of triangle ㄹㄴㄷ (DBC) in cm.
- All four sides of a square are equal.
- All radii in one circle are equal, so the diameter is twice the radius.
Plan
#1 Draw a Diagram · also uses: #7 Identify Subproblems
The picture carries the key facts: the diagonal is a diameter (so its length comes from the radius), and the triangle's other two sides are sides of the square. I find each side as a small subproblem, then add the three sides.
Execute
Review
The diagonal (14 cm) is the longest side, which matches the picture where the diameter stretches across the square. Two sides of 10 plus a 14 give 34 cm, a sensible perimeter that is larger than the square's side and smaller than the square's full perimeter.
Guess and check (tool 6): the square side must be 10 since 4 equal sides make 40, and the diameter must be 14 since two 7-cm radii line up along the diagonal; adding confirms 34 cm.
Standards · min grade 3
3.G.A.1Understand that shapes in different categories share attributes — Using that a square has four equal sides and that a line through the center is a diameter equal to twice the radius.3.MD.D.8Solve real-world problems involving perimeters of polygons — Adding the three side lengths to get the triangle's perimeter.