9 min readConcept sheet2-mark length question; 3-mark proof; part of a 5-mark question

Problems on tangents: circles inside triangles and quadrilaterals

When a circle sits inside a triangle or quadrilateral, each vertex sends two equal tangents. So for a circumscribed quadrilateral, AB + CD equals AD + BC.

Quadrilateral around a circleAB + CD = AD + BC (sums of opposite sides are equal)

Parallelogram around a circleIs a rhombus

Triangle around a circleEach side = sum of the two tangent lengths from its ends

Perimeter of such a triangle2 × (sum of the three different tangent lengths)

How to draw the figure

Draw a circle with centre O.
Draw a quadrilateral ABCD (or a triangle) whose every side just touches the circle.
Name the points of contact: P on AB, Q on BC, R on CD, S on DA.
Put matching tick marks on equal tangents: AP = AS, BP = BQ, CQ = CR, DR = DS.

Proof: AB + CD = AD + BC

The sides AB, BC, CD and DA of quadrilateral ABCD touch a circle at P, Q, R and S.

1★
Tangents from an external point are equal: AP = AS, BP = BQ, CR = CQ, DR = DS.
2
Add the four equations: AP + BP + CR + DR = AS + BQ + CQ + DS.
3
Group them: (AP + BP) + (CR + DR) = (AS + DS) + (BQ + CQ).
4
So AB + CD = AD + BC. Hence proved.

★ key step

Finding a side of a quadrilateral

Problem

A quadrilateral ABCD is drawn so that each side touches a circle. AB = 6 cm, BC = 7 cm and CD = 4 cm. Find AD.

Working

For a quadrilateral around a circle, AB + CD = AD + BC.
6 + 4 = AD + 7.
AD = 10 − 7 = 3 cm.

Answer

AD = 3 cm

Tangent lengths in a triangle

Problem

A circle touches the sides AB, BC and CA of ΔABC at P, Q and R. AB = 10 cm, BC = 12 cm and CA = 8 cm. Find AP, BQ and CR.

Working

Let AP = AR = x, BP = BQ = y and CQ = CR = z (equal tangents).
x + y = 10, y + z = 12, z + x = 8.
Add all three: 2(x + y + z) = 30, so x + y + z = 15.
x = 15 − 12 = 3, y = 15 − 8 = 7, z = 15 − 10 = 5.
Check: AB = 3 + 7 = 10, BC = 7 + 5 = 12, CA = 5 + 3 = 8.

Answer

AP = 3 cm, BQ = 7 cm, CR = 5 cm

Word problem: a fountain inside a plot

Problem

A garden plot is a parallelogram. A circular fountain inside it touches all four sides. One side of the plot is 9 m. Find the perimeter of the plot.

Working

The parallelogram circumscribes a circle, so AB + CD = AD + BC.
In a parallelogram, AB = CD and AD = BC. So 2AB = 2BC, which gives AB = BC.
All four sides are then equal, so the plot is a rhombus.
Perimeter = 4 × 9 = 36 m.

Answer

36 m

In real life

In a rangoli with a circle drawn inside a square or kite shape, the circle touches all four sides. The opposite sides then always add up to the same total.

Exam angle

2 marks: “A quadrilateral PQRS circumscribes a circle. PQ = 8 cm, QR = 9 cm and RS = 7 cm. Find SP.” PQ + RS = QR + SP, so 8 + 7 = 9 + SP and SP = 6 cm.

3 marks: “ABCD is a parallelogram whose four sides touch a circle. Show that ABCD is a rhombus.” AB + CD = AD + BC (equal tangents). With AB = CD and AD = BC, 2AB = 2BC, so AB = BC. All sides are equal: a rhombus.

3 marks: “A circle touches the sides BC, CA and AB of ΔABC at D, E and F. AB = 12 cm, BC = 8 cm and CA = 10 cm. Find AF, BD and CE.” Let AF = AE = x, BF = BD = y, CD = CE = z. x + y = 12, y + z = 8, z + x = 10, so x + y + z = 15. AF = 7 cm, BD = 5 cm, CE = 3 cm.

Remember it

Opposite sides, equal sums: AB + CD = AD + BC for any quadrilateral that hugs a circle.