Three sides, two arcs

The idea

Draw one given side as the base AB; from A draw an arc with the second length as radius and from B an arc with the third; their intersection C is at both distances, so it is the third vertex.

Where is the third corner?

We want a triangle with sides 4 cm, 5 cm and 6 cm. Draw the base AB = 4 cm. The third corner C must be 5 cm from A and 6 cm from B.

How can we find a point that is 5 cm from A and 6 cm from B?

• Keep trying with a ruler until both lengths fit
• Draw an arc with a compass from A and another from B
• Draw a line 5 cm long from A and a line 6 cm long from B

Draw the two arcs

Set the compass opening. Drag the pencil to draw the arcs. Watch how far the pencil is from A and from B.

The crossing is the corner

Every point on the arc from A is 5 cm from A. Every point on the arc from B is 6 cm from B.

Only the point where the arcs cross lies on both arcs. It is at both distances at once, so it is the third vertex C.

Draw one given side as the base AB. From A draw an arc with the second length as radius, and from B an arc with the third. Their intersection C is at both distances, so it is the third vertex.

• Step | What to do
• 1 | Draw the base AB = 4 cm.
• 2 | From A, draw an arc of radius 5 cm.
• 3 | From B, draw an arc of radius 6 cm. It crosses the first arc.
• 4 | Mark the crossing C. Join AC and BC.

Notes

Draw one given side as the base AB. From A draw an arc with the second length as radius and from B an arc with the third. Their intersection C is at both distances, so it is the third vertex.

Check yourself

A triangle has sides 3 cm, 5 cm and 6 cm. The base is AB = 6 cm and the arc from B has radius 5 cm. What radius should the arc from A have?

Why is the point where the arcs cross the third vertex of the triangle?

To build an equilateral triangle with side 4 cm on the base AB, what radius do you use for the arcs from A and from B?

Meena draws AB = 4 cm. Then she draws an arc of radius 5 cm from A and an arc of radius 5 cm from B. They cross at C. What is the perimeter of her triangle ABC (in cm)?

Answer: 14

AB = 4, AC = 5 and BC = 5, so the perimeter is 4 + 5 + 5 = 14 cm.

• 5 cm. The arc from B already uses 5 cm. The arc from A uses the remaining length.
• 6 cm. 6 cm is the base AB. It is drawn with the ruler. The arcs use the other two lengths.
• 3 cm — correct. Yes! The arcs use the two lengths that are not the base: 5 cm from B and 3 cm from A.
• It is the middle of the two arcs.. The crossing is not a middle point. It is the one point that lies on both arcs.
• It is on both arcs, so it is the right distance from A and from B. — correct. Yes! On the arc from A it is 5 cm from A. On the arc from B it is 6 cm from B.
• Both arcs are drawn with the same compass opening.. The openings can be different (5 cm and 6 cm here). What matters is that the point is on both arcs.
• 2 cm from A and 2 cm from B. Two arcs of 2 cm only touch in the middle of AB. They do not give a corner above the base.
• 4 cm from A and 4 cm from B — correct. Yes! The new corner must be 4 cm from A and 4 cm from B.
• 4 cm from A and 5 cm from B. Then BC would be 5 cm. All three sides must be 4 cm.