Triangle side equals sum of two radii

3.MD.D.83.G.A.1 · adapt · grade 3

Archetype: Radius and Diameter Relationships · step in a 11-type progression

In the figure on the right, the perimeter of triangle ABC is $44\ \text{cm}$ . What is the sum of the radii of the three circles, in centimeters?

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Understand

Three circles of different sizes each touch one another. Joining their centers makes triangle ABC. Each side equals the sum of the two touching circles' radii. The triangle's perimeter is 44 cm, and sides AB and AC are each 9 cm. I need the total of the three radii.

Givens

Three mutually touching circles; centers joined form triangle ABC (A top, B lower-left, C lower-right).
Each side equals the sum of the radii of the two circles meeting along it.
Perimeter of triangle ABC is 44 cm.
AB = 9 cm and AC = 9 cm.

Unknowns

The sum of the radii of the three circles, in cm.

Constraints

Two touching circles have center distance equal to the sum of their radii.
Each radius is counted in exactly two sides (each circle touches the other two).

Plan

#9 Solve an Easier Related Problem · also uses: #1 Draw a Diagram

Each side is a sum of two radii. Adding all three sides counts every radius twice, so the perimeter is double the radius total. The given AB and AC are not even needed for the total, which the radius-sum pattern makes clear.

Execute

#1 Draw a Diagram 3.G.A.1

Touching circles have center distance equal to the sum of their radii, so AB = rA + rB, AC = rA + rC, BC = rB + rC.

AB+AC+BC = (r_A+r_B)+(r_A+r_C)+(r_B+r_C)

Each triangle side bridges two circles that touch, so it is just their two radii laid end to end.

#9 Solve an Easier Related Problem 3.OA.D.9

Adding all three sides counts every radius exactly two times, because each circle touches the other two. So the perimeter equals twice the total of the three radii.

44 = 2 \times (r_A+r_B+r_C)

The repeated '+each radius twice' pattern turns the whole perimeter into a simple doubling.

#9 Solve an Easier Related Problem 3.OA.A.2

Since the perimeter is double the radius total, divide 44 by 2.

44 \div 2 = 22

Undoing a doubling is just dividing by 2.

Answer: 22 cm

Review

The radius total (22 cm) is exactly half the 44-cm perimeter, which is right because each radius is counted twice. As a check, BC = 44 - 9 - 9 = 26 cm, and 9 + 9 + 26 = 44 confirms the side lengths are consistent.

Find BC first: 44 - 9 - 9 = 26. Then add the three side equations another way, or guess radii like rA = 2, rB = 7, rC = 13 (sum 22) giving sides 9, 15, 20... adjust until AB = AC = 9; the radius total stays 22 regardless.

Standards · min grade 3

3.G.A.1 Understand that shapes in different categories share attributes — Recognizing each triangle side as the sum of two touching circles' radii.
3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Seeing that each radius is counted twice when adding the sides, so the perimeter is double the radius total.
3.OA.A.2 Interpret whole-number quotients of whole numbers — Dividing the perimeter by 2 to find the sum of the radii.

💡 This only needs Grade 3 thinking: each radius shows up twice around the triangle, so halve the perimeter!