Triangles

Fill in the blanks using the correct word given in brackets :

(i) All circles are __________ . (congruent, similar)

(ii) All squares are __________ . (similar, congruent)

(iii) All __________ triangles are similar. (isosceles, equilateral)

(iv) Two polygons of the same number of sides are similar, if (a) their corresponding angles are __________ and (b) their corresponding sides are __________ . (equal, proportional)

Give two different examples of pair of

(i) similar figures. (ii) non-similar figures.

State whether the following quadrilaterals are similar or not:

In Fig. 6.17, (i) and (ii), $DE \parallel BC$. Find $EC$ in (i) and $AD$ in (ii).

$E$ and $F$ are points on the sides $PQ$ and $PR$ respectively of a $\Delta PQR$. For each of the following cases, state whether $EF \parallel QR$ :

(i) $PE = 3.9\text{ cm}$, $EQ = 3\text{ cm}$, $PF = 3.6\text{ cm}$, and $FR = 2.4\text{ cm}$

(ii) $PE = 4\text{ cm}$, $QE = 4.5\text{ cm}$, $PF = 8\text{ cm}$, and $RF = 9\text{ cm}$

(iii) $PQ = 1.28\text{ cm}$, $PR = 2.56\text{ cm}$, $PE = 0.18\text{ cm}$, and $PF = 0.36\text{ cm}$

In Fig. 6.18, if $\text{LM} \parallel \text{CB}$ and $\text{LN} \parallel \text{CD}$, prove that

$\dfrac{\text{AM}}{\text{AB}} = \dfrac{\text{AN}}{\text{AD}}$

In Fig. 6.19, $DE \parallel AC$ and $DF \parallel AE$. Prove that

$\dfrac{\text{BF}}{\text{FE}} = \dfrac{\text{BE}}{\text{EC}}$

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 :

2. In Fig. 6.35, $\Delta \text{ODC} \sim \Delta \text{OBA}$, $\angle \text{BOC} = 125^\circ$ and $\angle \text{CDO} = 70^\circ$. Find $\angle \text{DOC}$, $\angle \text{DCO}$ and $\angle \text{OAB}$.

3. Diagonals $\text{AC}$ and $\text{BD}$ of a trapezium $\text{ABCD}$ with $\text{AB} \parallel \text{DC}$ intersect each other at the point $\text{O}$. Using a similarity criterion for two triangles, show that $\dfrac{\text{OA}}{\text{OC}} = \dfrac{\text{OB}}{\text{OD}}$.

4. In Fig. 6.36, $\dfrac{\text{QR}}{\text{QS}} = \dfrac{\text{QT}}{\text{PR}}$ and $\angle 1 = \angle 2$. Show that $\Delta \text{PQS} \sim \Delta \text{TQR}$.

S and T are points on sides $PR$ and $QR$ of $\Delta \text{PQR}$ such that $\angle \text{P} = \angle \text{RTS}$. Show that $\Delta \text{RPQ} \sim \Delta \text{RTS}$.

6. In Fig. 6.37, if $\Delta \text{ABE} \cong \Delta \text{ACD}$, show that $\Delta \text{ADE} \sim \Delta \text{ABC}$.

7. In Fig. 6.38, altitudes $AD$ and $CE$ of $\Delta \text{ABC}$ intersect each other at the point $P$. Show that:

(i) $\Delta \text{AEP} \sim \Delta \text{CDP}$

(ii) $\Delta \text{ABD} \sim \Delta \text{CBE}$

(iii) $\Delta \text{AEP} \sim \Delta \text{ADB}$

(iv) $\Delta \text{PDC} \sim \Delta \text{BEC}$

8. E is a point on the side AD produced of a parallelogram ABCD and BE intersects CD at F. Show that $\Delta \text{ABE} \sim \Delta \text{CFB}$.

9. In Fig. 6.39, ABC and AMP are two right triangles, right angled at B and M respectively. Prove that:

(i) $\Delta \text{ABC} \sim \Delta \text{AMP}$

(ii) $\dfrac{\text{CA}}{\text{PA}} = \dfrac{\text{BC}}{\text{MP}}$

10. CD and GH are respectively the bisectors of $\angle \text{ACB}$ and $\angle \text{EGF}$ such that D and H lie on sides AB and FE of $\Delta \text{ABC}$ and $\Delta \text{EFG}$ respectively. If $\Delta \text{ABC} \sim \Delta \text{FEG}$, show that:

(i) $\dfrac{\text{CD}}{\text{GH}} = \dfrac{\text{AC}}{\text{FG}}$

(ii) $\Delta \text{DCB} \sim \Delta \text{HGE}$

(iii) $\Delta \text{DCA} \sim \Delta \text{HGF}$

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 $\Delta ABD \sim \Delta ECF$.

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

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

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 $\Delta ABC \sim \Delta PQR$.

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

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

$\dfrac{\text{AB}}{\text{PQ}} = \dfrac{\text{AD}}{\text{PM}}$

Draw two line segments $BC$ and $EF$ of two different lengths, say $3\text{ cm}$ and $5\text{ cm}$ respectively. Then, at the points $B$ and $C$ respectively, construct angles $\angle PBC$ and $\angle QCB$ of some measures, say, $60^\circ$ and $40^\circ$. Also, at the points $E$ and $F$, construct angles $\angle REF$ and $\angle SFE$ of $60^\circ$ and $40^\circ$ respectively (see Fig. 6.23).