Grade 12 Maths · Analytical Geometry · Find the length of a tangent

Find the length of a tangent Grade 12

Find the length of a tangent from a point outside a circle by using the distance from the point to the centre and the radius in Pythagoras, for example distance 13 and radius 5 give a tangent of 12.

questions use it, all grades · 2 past paper, 10 practice: 12 questions use it, all grades · 2 past paper, 10 practice
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Analytical Geometry: Lesson: Length of a Tangent to a CircleMindset

Circle - Length of Tangent from External PointDeasy Maths

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Practice · NavyBlue · 3 marks

A circle with centre

M

has equation

x^{2} + y^{2} + 2 x - 4 y - 20 = 0

. The point

A (3; 5)

lies on the circle.

The point

T (9; - 6)

lies outside the circle. Calculate the length of the tangent from

T

to the circle.(3)

Work it out on paper first. Then mark yourself with the memo, one tick at a time.

More questions: find the length of a tangent

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3. Question 4.4.1 · November 2022 · Paper 2 · National (DBE) · 2 marks
In the diagram, $M (3; - 5)$ is the centre of the circle having PN as its diameter. KL is a tangent to the circle at $N (7; - 2)$ . $Diagram for this question$
4.4 Points $A (t; t)$ and B are not shown on the diagram.

From point A, another tangent is drawn to touch the circle with centre M at B.

Answer the questions that follow.
4.4.1 Show that the length of tangent AB is given by $2 t^{2} + 4 t + 9$ .(2)
Show the memo (2 ticks)Memo
1. ✓
  Substitution into Pythagoras $A B^{2} = A M^{2} - M B^{2}$
  $A B^{2} = [(t - 3)^{2} + (t + 5)^{2}] - 5^{2}$
  1 mark · Find the length of a tangent
2. ✓
  Simplification (A) $= t^{2} - 6 t + 9 + t^{2} + 10 t + 25 - 25$
  $A B = 2 t^{2} + 4 t + 9$
  1 mark · Square a binomial
4. Question 4.1.5 · September 2024 · Paper 2 · Provincial · 5 marks
4.1 The diagram below shows two circles touching at $J (2; - 1)$ .
The smaller circle has its centre at the origin and the bigger circle has centre $K (a; - 3)$ .
The length of the radius of the bigger circle is TWICE the length of the radius of the smaller circle.
SR is a tangent to both circles, touching the bigger circle at S and the smaller circle at R.
KO and SR are both produced to intersect in point P.

Answer the questions that follow. $Diagram for question 4.1$
4.1.5 Calculate the length of PS.(5)
Show the memo (5 ticks)Memo
1. ✓
  Length of KO $K O = 5 + 25 = 35$
  1 mark · Decide whether two circles touch
2. ✓
  $PO = \frac{1}{2} P K$ In $△$ POR and $△$ PKS:
  $\hat{P} = \hat{P}$ [common]
  $P \hat{R} O = P \hat{S} K$ [ $= 90°$ ; tangent $⊥$ radius]
  $P \hat{O} R = P \hat{K} S$ [remaining $∠$ s]
  $△ POR ∣∣∣ △ P K S$ [ $∠∠∠$ ]
  $\frac{PO}{P K} = \frac{OR}{K S}$ [ $∣∣∣ △$ s]
  $= \frac{OR}{2 OR} = \frac{1}{2}$
  $∴ PO = \frac{1}{2} P K$
  1 mark · Use similar triangles' sides and angles
3. ✓
  Length of PK $PO = O K = 35$
  $P K = 2 (35) = 65$
  1 mark · Use similar triangles' sides and angles
4. ✓
  Substitution in Theorem of Pythagoras $P \hat{S} K = 90°$ [radius $⊥$ tangent]
  $P S^{2} = P K^{2} - K S^{2}$ [Theorem of Pythagoras]
  $= (65)^{2} - (25)^{2}$
  1 mark · Find the length of a tangent
5. ✓
  Answer $= 160$
  $∴ PS = 160 = 410$
  1 mark · Use the theorem of Pythagoras
5. Practice · NavyBlue · 3 marks
$A (- 3; 1)$ , $B (3; - 1)$ and $C (- 1; 3)$ lie on the circle $x^{2} + y^{2} = 10$ with centre $O$ .
Calculate the length of the tangent from $K (5; 5)$ to the circle.(3)
Show the memo (3 ticks)Memo
1. ✓
  OK squared $O K^{2} = 5^{2} + 5^{2} = 50$
  1 mark · Calculate distance between two points
2. ✓
  Pythagoras $(tangent)^{2} = 50 - 10 = 40$
  1 mark · Find the length of a tangent
3. ✓
  LengthTangent $= 40 = 210 \approx 6, 32$ units
  1 mark · Find the length of a tangent
6. Practice · NavyBlue · 2 marks
The circle with centre $M$ has equation $(x - 1)^{2} + (y - 2)^{2} = 20$ . $PT$ is a tangent to the circle at $T$ , where $P$ is $(7; 10)$ .
(b) Calculate the length of $PT$ in simplest surd form.(2)
Show the memo (2 ticks)Memo
1. ✓
  Pythagoras $P T^{2} = P M^{2} - M T^{2} = 100 - 20 = 80$ (tan $⊥$ radius)
  1 mark · Find the length of a tangent
2. ✓
  Simplify $PT = 80 = 45$
  1 mark · Simplify and combine surds

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