Triangles | Exercise 6.3

Question 11

In Fig. 6.40, E is a point on side CB produced of an isosceles triangle ABC with AB = AC. If AD ⊥\perp BC and EF ⊥\perp AC, prove that Δ\Delta ABD ∼\sim Δ\Delta ECF.

Question diagram 1
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Solution

We will use the following similarity criterion:

  • AA (Angle-Angle): If two angles of one triangle are equal to two corresponding angles of another triangle, the triangles are similar.

We need to prove that Δ\Delta ABD is similar to Δ\Delta ECF.

Step 1 — Identify equal angles from isosceles triangle

Triangle ABC is an isosceles triangle. Its sides AB and AC are equal. Angles opposite to equal sides are equal. So, ∠\angle ABC equals ∠\angle ACB. We can write ∠\angle ABD for ∠\angle ABC. We can write ∠\angle ECF for ∠\angle ACB. Therefore, ∠\angle ABD = ∠\angle ECF.

Step 2 — Identify right angles

AD is perpendicular to BC. This means ∠\angle ADB is a right angle. So, ∠\angle ADB = 90°. EF is perpendicular to AC. This means ∠\angle EFC is a right angle. So, ∠\angle EFC = 90°.

Step 3 — Apply AA similarity criterion

Let's compare Δ\Delta ABD and Δ\Delta ECF. From Step 2, we know ∠\angle ADB = ∠\angle EFC. Both these angles are 90°. From Step 1, we know ∠\angle ABD = ∠\angle ECF. These are two pairs of equal angles. By the AA similarity criterion, the triangles are similar.

Δ ABD ∼Δ ECF\boxed{\Delta \text{ ABD } \sim \Delta \text{ ECF}}

Diagram 1

More questions in Exercise 6.3

Q1
  1. State which pairs of triangles in Fig. 6.34 are similar. Write the similarity criterion used by you for answering the question and also write the pairs of similar triangles in the symbolic form :
Q2
  1. In Fig. 6.35, ΔODC∼ΔOBA\Delta \text{ODC} \sim \Delta \text{OBA}, ∠BOC=125∘\angle \text{BOC} = 125^\circ and ∠CDO=70∘\angle \text{CDO} = 70^\circ. Find ∠DOC\angle \text{DOC}, ∠DCO\angle \text{DCO} and ∠OAB\angle \text{OAB}.
Q3
  1. Diagonals AC and BD of a trapezium ABCD with AB∥DC\text{AB} \parallel \text{DC} intersect each other at the point O. Using a similarity criterion for two triangles, show that OAOC=OBOD\frac{\text{OA}}{\text{OC}} = \frac{\text{OB}}{\text{OD}}.
Q4
  1. In Fig. 6.36, QRQS=QTPR\frac{\text{QR}}{\text{QS}} = \frac{\text{QT}}{\text{PR}} and ∠1=∠2\angle 1 = \angle 2. Show that ΔPQS∼ΔTQR\Delta \text{PQS} \sim \Delta \text{TQR}.
Q5
  1. S and T are points on sides PR and QR of ΔPQR\Delta \text{PQR} such that ∠P=∠RTS\angle \text{P} = \angle \text{RTS}. Show that ΔRPQ∼ΔRTS\Delta \text{RPQ} \sim \Delta \text{RTS}.
Q6
  1. In Fig. 6.37, if ΔABE≅ΔACD\Delta \text{ABE} \cong \Delta \text{ACD}, show that ΔADE∼ΔABC\Delta \text{ADE} \sim \Delta \text{ABC}.
Q7
  1. In Fig. 6.38, altitudes AD and CE of ΔABC\Delta \text{ABC} intersect each other at the point P. Show that:

(i) ΔAEP∼ΔCDP\Delta \text{AEP} \sim \Delta \text{CDP} (ii) ΔABD∼ΔCBE\Delta \text{ABD} \sim \Delta \text{CBE} (iii) ΔAEP∼ΔADB\Delta \text{AEP} \sim \Delta \text{ADB} (iv) ΔPDC∼ΔBEC\Delta \text{PDC} \sim \Delta \text{BEC}

Q8
  1. E is a point on the side AD produced of a parallelogram ABCD and BE intersects CD at F. Show that ΔABE∼ΔCFB\Delta \text{ABE} \sim \Delta \text{CFB}.
Q9
  1. In Fig. 6.39, ABC and AMP are two right triangles, right angled at B and M respectively. Prove that:

(i) ΔABC∼ΔAMP\Delta \text{ABC} \sim \Delta \text{AMP} (ii) CAPA=BCMP\frac{\text{CA}}{\text{PA}} = \frac{\text{BC}}{\text{MP}}

Q10
  1. CD and GH are respectively the bisectors of ∠ACB\angle \text{ACB} and ∠EGF\angle \text{EGF} such that D and H lie on sides AB and FE of ΔABC\Delta \text{ABC} and ΔEFG\Delta \text{EFG} respectively. If ΔABC∼ΔFEG\Delta \text{ABC} \sim \Delta \text{FEG}, show that:

(i) CDGH=ACFG\frac{\text{CD}}{\text{GH}} = \frac{\text{AC}}{\text{FG}} (ii) ΔDCB∼ΔHGE\Delta \text{DCB} \sim \Delta \text{HGE} (iii) ΔDCA∼ΔHGF\Delta \text{DCA} \sim \Delta \text{HGF}

Q11

In Fig. 6.40, E is a point on side CB produced of an isosceles triangle ABC with AB = AC. If AD ⊥\perp BC and EF ⊥\perp AC, prove that Δ\Delta ABD ∼\sim Δ\Delta ECF.

Q12

Sides AB and BC and median AD of a triangle ABC are respectively proportional to sides PQ and QR and median PM of Δ\Delta PQR (see Fig. 6.41). Show that Δ\Delta ABC ∼\sim Δ\Delta PQR.

Q13

D is a point on the side BC of a triangle ABC such that ∠\angle ADC = ∠\angle BAC. Show that CA2=CB.CD\text{CA}^2 = \text{CB.CD}.

Q14

Sides AB and AC and median AD of a triangle ABC are respectively proportional to sides PQ and PR and median PM of another triangle PQR. Show that Δ\Delta ABC ∼\sim Δ\Delta PQR.

Q15

A vertical pole of length 6 m casts a shadow 4 m long on the ground and at the same time a tower casts a shadow 28 m long. Find the height of the tower.

Q16

If AD and PM are medians of triangles ABC and PQR, respectively where Δ\Delta ABC ∼\sim Δ\Delta PQR, prove that

ABPQ=ADPM\frac{\text{AB}}{\text{PQ}} = \frac{\text{AD}}{\text{PM}}

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