Grade 12 Maths · Euclidean Geometry · Use the altitude in a right triangle

Use the altitude in a right triangle Grade 12

Use the three similar triangles formed when a line is drawn from the right angle perpendicular to the hypotenuse, for example AD^2 = BD DC, to find lengths or prove results.

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EUCLIDEAN GEOMETRY | GRADE 12| SIMILAR RIGHT TRIANGLEBACK2BASICS LEARNING INSTITUTE

Altitude on Hypotenuse Theorem - Geometry Practice ProblemsThe Organic Chemistry Tutor

How to determine the similarity of three triangles by drawing the altitudeBrian McLogan

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Practice · NavyBlue · 2 marks

Δ A BC

B \hat{A} C = 9 0^{\circ}

and

A D ⊥ BC

B D = 4

and

D C = 9

(a) Complete:

Δ A B D ∣∣∣ Δ \dots ∣∣∣ Δ \dots

(2)

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More questions: use the altitude in a right triangle

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3. Question 10.2.2(a) · September 2025 · Paper 2 · Provincial · 4 marks
10.2 In the diagram below A, C and F are points on a semi-circle with centre B. A circle with centre C is drawn. The radius of the semi-circle is twice the radius of circle C.
FED is a tangent to both circles at F and E respectively. ABCD is a straight line.

Answer the questions that follow. $Diagram for question 10.2$
10.2.2 If the radius of the smaller circle is $p$ , determine the following in terms of $p$ :
10.2.2(a) The length of FC(4)
Show the memo (4 ticks)Memo
1. ✓
  AC = 4pEC $= p$ , BC $= 2 p$
  AC $= 4 p$
  1 mark · Use radius and tangent facts
2. ✓
  S/R $\frac{FC}{EC} = \frac{AC}{FC}$ ( $△$ AFC $∣∣∣$ $△$ FEC)
  1 mark · Use similar triangles' sides and angles
3. ✓
  SubstitutionFC $^{2} =$ AC.EC
  $∴$ FC $^{2} = 4 p \times p$
  1 mark · Use the altitude in a right triangle
4. ✓
  AnswerFC $= 4 p^{2}$
  $∴$ FC $= 2 p$
  1 mark · Use similar triangles' sides and angles
4. Practice · NavyBlue · 2 marks
In $Δ PQR$ , $Q \hat{P} R = 9 0^{\circ}$ and $PS ⊥ QR$ with $S$ on $QR$ .
If $QS = 5$ and $PS = 10$ , calculate $SR$ and $PQ$ .(2)
Show the memo (2 ticks)Memo
1. ✓
  SR $100 = 5 \cdot SR$ , so $SR = 20$
  1 mark · Use the altitude in a right triangle
2. ✓
  PQ $P Q^{2} = QS \cdot QR = 5 \times 25 = 125$ , so $PQ = 55$
  1 mark · Use the altitude in a right triangle
5. Practice · NavyBlue · 3 marks
$A B$ is a diameter of the circle with centre $O$ . $C$ lies on the circle and $C D ⊥ A B$ with $D$ on $A B$ . $A D = 2$ and $D B = 8$ .
(b) Prove that $C D^{2} = A D \cdot D B$ .(3)
Show the memo (3 ticks)Memo
1. ✓
  Statement and reason $Δ A C D ∣∣∣ Δ CB D$ [ $∠∠∠$ ]
  2 marks · Use the altitude in a right triangle
2. ✓
  Ratio $\frac{C D}{B D} = \frac{A D}{C D}$ , so $C D^{2} = A D \cdot D B$
  1 mark · Use the altitude in a right triangle
6. Practice · NavyBlue · 2 marks
$A B$ is a diameter of the circle with centre $O$ . $C$ is a point on the circle and $C D ⊥ A B$ with $D$ on $A B$ . $A D = 2$ and $D B = 8$ .
Calculate $C D$ .(2)
Show the memo (2 ticks)Memo
1. ✓
  Altitude result $C D^{2} = A D \cdot D B$ ( $Δ A C D ∣∣∣ Δ CB D$ )
  1 mark · Use the altitude in a right triangle
2. ✓
  CD $C D^{2} = 2 \times 8 = 16$ , $C D = 4$
  1 mark · Use the altitude in a right triangle

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