# Similar Triangles (Myanmar Exam Board)

Group (2015-2019)

1. (2015/Myanmar /q13a )
In $\triangle A B C, A D$ and $B E$ are the altitudes. If $\alpha(\triangle D E C)=\frac{3}{4} \alpha^{\prime} \cdot{ }^{\prime} \mathrm{C}$ ), prove that $\angle A C B=30^{\circ}, \quad(5$ marks $)$

2. (2015/FC /q13a )
$A B C$ is a triangle. If $B P C, C Q A, A R B$ are equilateral triangles and $\alpha(\triangle B P C)+\alpha(\Delta C Q A)=\alpha(\triangle A R B)$, then prove that $A B C$ is a right triangle. $(5$ marks $)$

3. (2016/Myanmar /q13a )
In the figure, $A B / / C D$ and $\alpha(\triangle E C D): \alpha(A B D C)=16: 9$. Find the numerical value of $C D: A B$.

4. (2016/FC /q13a )
$A, B, C$ and $D$ are four points in order on a circle $O$, so that $A B$ is a diameter and $\angle C O D=90^{\circ}$. If $A D$ produced and $B C$ produced meet at $E$, prove that $\alpha(\Delta E C D)=\alpha(A B C D)$

6. (2017/Myanmar /q13a )
In $\triangle A B C, A D$ and $B E$ are altitudes to the sides $B C$ and $A C$ respectively. If $\angle A C D=45^{\circ}$, prove that $\alpha(\triangle D E C): \alpha(\triangle A B C)=1: 2$
Q13(a) Solution

7. (2017/FC /q13a )
In trapezium $\mathrm{ABCD}$ the diagonals $\mathrm{AC}$ and $\mathrm{BD}$ intersect at $\mathrm{O}$. If $\mathrm{AB} / / \mathrm{DC}$ and $16 \alpha(\triangle \mathrm{AOB})=25 \alpha(\Delta \mathrm{COD})$, find the ratios $\mathrm{AB}: \mathrm{CD}$ and $\alpha(\triangle \mathrm{BOC}): \alpha(\Delta \mathrm{COD})$ $(5 \mathrm{marks})$

8. (2018/Myanmar /q13a )
$A B C$ is a right triangle with $\angle A$ the right angle. $E$ and $D$ are points on opposiie side of $A C$, with $E$ on the same side of $A C$ as $B$, such that $\triangle A C D$ and $\Delta B C E$ are both equilateral. If $\alpha(\triangle B C E)=2 \alpha(\triangle A C D)$, Prove that $A B C$ is an isosceles right triangle.

Click for Solution 9. (2018/FC /q13a )
In the figure $\angle A P Q=\angle C, A P: P B=3: 1$ and $A Q: Q C=1: 2$. If $A Q=2$, find the length of $\mathrm{AP}$ and the ratios of $\alpha(\triangle A P Q): \alpha(\triangle A B C)$ and $\alpha(\triangle A P Q): \alpha(B C Q P)$.

10. (2019/Myanmar /q11b )
In the diagram, $\mathrm{P}$ is the point on $\mathrm{AC}$ such that $\mathrm{AP}=3 \mathrm{PC}, \mathrm{R}$ is the point on BP such that $\mathrm{BR}=2 \mathrm{RP}$ and $\mathrm{QR} / / \mathrm{AC}$. Given that $\alpha(\triangle \mathrm{APA})=36 \mathrm{~cm}^{2}$, calculate $\alpha(\Delta \mathrm{BPC})$ and $\alpha(\Delta \mathrm{BRQ}), \quad \mathrm{Q} \quad \mathrm{A} \quad(5 \mathrm{marks})$ Click for Solution

11. (2019/FC /q11b )
Given : $\triangle P Q R$ with two medians $P M$ and $\mathrm{QN}$ intersecting at $\mathrm{K}$. $\quad / \mathrm{K} / \mathrm{N} \quad$ (5 marks) Prove $: \alpha(\triangle \mathrm{PNK})=\alpha(\Delta \mathrm{QMK})$ Click for Solution 11(b)

1.  Prove
2.  Prove
3.  $\alpha(\Delta E A B)=37.5 \mathrm{~cm}^{2}$
4.  Prove
6.  Prove
7.  $\dfrac{5}{4},5:4$
8.   Prove
9.   $A P=3, \frac{1}{4}, \frac{1}{3}$
10.   16
11.  Prove

Group (2014)

1. $A, B, C, D$ are four points in order on a circle $O$, so that $A B$ is a diameter. $A D$ and $B C$ meet at $E$. If $\alpha(A B C D)=3 \alpha(\triangle E C D)$, prove that $\angle C O D=60^{\circ}$. (5 marks)

2. $A, B, C, D$ are four points in order on a circle $O$, so that $A B$ is a diameter. $A D$ and $B C$ meet at $\mathrm{E} .$ If $\alpha(\triangle E C D)=\alpha(A B C D)$, prove that $D C=\sqrt{2} A O$. (5 marks)

3. In $\triangle A B C, A D \perp B C$ and $B E \perp A C$. If $2 \alpha(\triangle D E C)=\alpha(\triangle A B C)$, find $\angle A C B$. (5 marks)

4. In $\triangle A B C, \angle B A C=90^{\circ}$ and $\angle A B C=30^{\circ} . D$ and $E$ are points on opposites of $A C$, with $E$ on the same side of $A C$ as $B$, such that both $\triangle A C D$ and $\triangle B C E$ are equilateral. Prove that $\alpha(\triangle B C E)=4 \alpha(\triangle A C D)$. (5 marks)

5. In $\triangle A B C, A D$ and $B E$ are altitudes. If $\angle A C B=45^{\circ}$, prove that $\alpha(\Delta D E C)=\alpha(A B D E)$. (5 marks)

6. $\triangle A B C, A D$ and $B E$ are altitudes. If $\angle A C B=30^{\circ}$, prove that $\alpha(\triangle D E C)=3 \alpha(A B D E)$. (5 marks)

7. $P Q R S$ is a parallelogram. $P S$ is produced to $L$ so that $S L=S R$ and $L R$ produced meets $P Q$ produced at $M$. Prove that $Q M=Q R$. If the area of the parallelogram is $20 \mathrm{~cm}^{2}$ and $P Q=2 P S$, find the area of $\triangle L S R$. (5 marks)

8. In $\triangle A B C, D$ is a point of $A C$ such that $A D=3 C D . \mathrm{E}$ is on $B C$ such that $D E / / A B$. Compare the areas of $\triangle C D E$ and $\triangle A B C$. If $\alpha(A B E D)=30$, what is $\alpha(\Delta A B C) ?$ (5 marks)

9. The area of $\triangle A B C$ is bisected by a line $P Q$ drawn parallel to $B C$ where $P$ lies on the side $A B$ and $Q$ lies on the side $A C$. In what ratio does $P Q$ divide $A B$ and $A C$ ? Find also the ratio $\alpha(\triangle B P Q): \alpha(\triangle B C Q)$. (5 marks)

10. In the diagram, $P$ is the point on $A C$ such that $A P=3 P C, R$ is the point on $B P$ such that $B P=3 R P$ and $Q R \| A C$. Given that $\alpha(\triangle B P A)=36 \mathrm{~cm}^{2}$, calculate the areas of $\triangle B P C$ and $\triangle B R Q .$ (5 marks)

1. Prove
2. Prove
3. $\angle A C B=45^{\circ}$
4. Prove
5. Prove
6. Prove
7. $20 \mathrm{~cm}^{2}$
8. $1: 16 / 32$
9. $1:(\sqrt{2}-1), 1: \sqrt{2}$
10. $12 \mathrm{~cm}^{2}, 16 \mathrm{~cm}^{2}$

Group (2013)

1. In trapezium $A B C D, A B$ is twice $D C$ and $A B \| D C$. If $A C, B D$ intersect at $O$, prove that $\alpha(\triangle A O B)=4 \alpha(\triangle C O D)$. Find the ratio of $A O: C O$. (5 marks)

2. In $\triangle A B C, \angle A=90^{\circ}, A C=5$ and $B C=13 . D$ is a point on $A B$ such that $D E \perp B C$ and $C E=5$. Find $\alpha(\triangle A B C): \alpha(\triangle B D E)$ and find $\alpha(A D E C)$ if $\alpha(\triangle \mathrm{BDE})=\frac{40}{3}$. (5 marks)

3. In $\triangle A B C, \angle B A C=90^{\circ}$ and $A D \perp B C$. If $D C=8 B D$, prove that $B C=3 A B$. (5 marks)

4. In $\triangle A B X, C$ is a point on the segment $B X$ and $D$ is a point on the segment $A X$ such that $\angle B A C=\angle B D C$. Prove that $\alpha(\triangle A B X): \alpha(\Delta C D X)=A B^{2}: C D^{2}$. (5 marks)

5. $A B C D$ is a segment and $P$ a point outside it such that $\angle P B A=\angle P C D=\angle A P D$. Prove that $\frac{\alpha(\triangle A B P)}{\alpha(\Delta P C D)}=\frac{A B^{2}}{B P^{2}}$. (5 marks)

6. A quadrilateral $A B C D$ is inscribed in a circle. $A B$ and $D C$ are produced to meet at $E$. If $A D=6 \mathrm{~cm}, B C=4 \mathrm{~cm}$ and the area of triangle $B C E$ is $12 \mathrm{~cm}^{2}$, calculate the area of $A B C D$. (5 marks)

7. Two straight lines $A B$ and $C D$ intersect at $E . C M \perp A E, D N \perp B E$ and $\frac{E A}{E C}=\frac{E B}{E D} .$ If $C M=3, D N=4$ and $A B=14$, find $\frac{\alpha(\triangle A C E)}{\alpha(\Delta B D E)}$ and $\alpha(\triangle A C E)$. (5 marks)

8. $\triangle A B C$ is an isosceles right triangle with $A$ the right angle. $E$ and $D$ are points on opposite side of $A C$, with $E$ on the same side of $A C$ as $B$, such that $A C D$ and $B C E$ are both equilateral. Prove that $\alpha(\triangle B C E)=2 \alpha(\triangle A C D) .$ (5 marks)

9. In $\triangle P Q R, Q R=32 \mathrm{~cm}$. The point $Y$ on $P R$ is such that $P Y=6 \mathrm{~cm}, Y R=10 \mathrm{~cm}$. The point $X$ on $P Q$ is such that $X Y / / Q R$. Find the length of $X Y$. If $\alpha(\Delta P X Y)=27 \mathrm{~cm}^{2}$, find $\alpha(Q X Y R)$. (5 marks)

10. In a $\triangle A B C$ the side $A B$ is divided at $X$ so that $\frac{A X}{X B}=\frac{3}{2} .$ A line through $X$ parallel to $B C$, meets $A C$ at $Y$. Find $\alpha(\triangle A B C): \alpha(\triangle B C Y)$. (5 marks)

11. In $\triangle A B C, A D$ and $B E$ are the altitudes. If $\angle A C B=30^{\circ}$, prove that $\alpha(\triangle D E C)=\frac{3}{4} \alpha(\Delta A B C)$. (5 marks)

1. $AO:CO=2:1$
2. $\alpha(\triangle ABC):\alpha(\triangle BDE)=9:4$
3. Prove
4. Prove
5. Prove
6. Prove
7. Prove
8. Prove
9. Prove
10. Prove
11. Prove

$\qquad$$\, 1.A B C is a triangle. If B P C, C Q A, A R B are equilateral triangles, and \alpha(\triangle B P C)+\alpha(\Delta C Q A)=\alpha(\triangle A R B) then prove that A B C is a right triangle. (5 marks) 2.A, B, C, D are four points in order on a circle O, so that A B is a diameter and \angle C O D=90^{\circ} . A D and B C meet at E, prove that \alpha(\triangle E C D)=\alpha(A B C D). (5 marks) 3.A B C is a triangle such that B C: C A: A B=3: 4: 5. If B P C, C Q A, A R B are equilateral triangles, prove that \alpha(\triangle B P C)+\alpha(\Delta C Q A)=\alpha(\triangle A R B). (5 marks) 4.A, B, C and D are four points in order on a circle O so that A B is a diameter and \angle C O D=90^{\circ}. If A D produced and B C produced meet at E, prove that \alpha(\Delta E C D)=\alpha(A B C D). (5 marks) 5.In \triangle P Q R, P S and Q T are the altitudes. If \angle P R Q=60^{\circ}, then prove that 4 \alpha(\Delta S T R)=\alpha(\triangle P Q R). (5 marks) 6.(figure) A D P and B C P are two segments such that \angle B A C=\angle B D C . Prove that \frac{\alpha(\triangle B A P)}{\alpha(\triangle C D P)}=\frac{A B^{2}}{C D^{2}}. (5 marks) 7.A B C D is a trapezium in which A B / / C D and \angle A D B=\angle C. Prove that A D^{2}: B C^{2}=A B: C D. (5 marks) 8.Two chords A C and B D of a circle intersect at O. Show that$$\alpha(\triangle A O B): \alpha(\Delta C O D)=O A^{2}: O D^{2}$$Show also that$$\alpha(\triangle A O B): \alpha(\triangle A O D)=O B: O D$$. (5 marks) 9.A B C D is a parallelogram and P Q / / B M, where Q is midpoint of C D and P, M are points on B C and A D respectively. If \alpha(\triangle P C Q)=25 \mathrm{~cm}^{2}, find \alpha(\triangle A B M). (5 marks) 10.P, Q, R and S are four points in order on a circle O, such that P Q is a diameter. P S and Q R meet at T. If \alpha(\Delta T R S)=\alpha(P Q R S), show that R S=\sqrt{2} O P. (5 marks) 11.In \triangle A B C, \angle A=90^{\circ} and A S \perp B C. If 2 B C=3 A B, find the ratio of B S: C S. (5 marks) 12.The sides A B and B C of \triangle A B C are 5 \mathrm{~cm} and 6 \mathrm{~cm} respectively. Points H and K on A B and A C respectively are such that H K and B C are parallel. If the areas of triangles A H K and A B C are in the ratio of 4: 9 ; calculate H K and H B. (5 marks) #### Answer (2012) \quad \, 1.Prove 2.Prove 3.Prove 4.Prove 5.Prove 6.Prove 7.Prove 8.Prove 9.100 \mathrm{~cm}^{2} 10.Prove 11.4: 5 12.4 \mathrm{~cm} ; \frac{5}{3} \mathrm{~cm} ## Group (2011) \quad\;\,$$\,$
1.Two chords $X B$ and $A Y$ of a circle intersect at $S .$ If $X S=4 \mathrm{~cm}, S A=5 \mathrm{~cm}$, then prove that $\triangle X Y S \sim \triangle A B S$, and hence, find $\alpha(\Delta X Y S): \alpha(\triangle A B S)$. $\mbox{ (5 marks)}$

2.The chords $B X$ and $A Y$ of a circle intersect at $S .$ If $B S=3 \mathrm{~cm}, A S=5 \mathrm{~cm}$ and $X S=4 \mathrm{~cm}$, find $\alpha(\triangle A B S): \alpha(\Delta X Y S) .$ $\mbox{ (5 marks)}$

3.In the diagram, $R$ is the point on $B P$ such that $B R=2 R P$ and $Q R / I A C .$ Given that $\alpha(\triangle B P A)=18 \mathrm{~cm}^{2}$ calculate $\alpha(\Delta B R Q), \alpha(P A Q R)$. $\mbox{ (5 marks)}$

4.$\quad A B C$ is a right triangle with $\angle A$ the right angle.$E$ and $D$ are points onc $+$ posite side of $A C$, with $E$ on the same side of $A C$ as $B$, such that $\triangle A C D$ and $\triangle B C E$ are both equilateral.If $\alpha(\triangle B C E)=2 \alpha(\triangle A C D)$, prove that $A B C$ is an isosceles right triangle. $\mbox{ (5 marks)}$

5.$\triangle A B C$ is an isosceles right triangle with $\angle A$ the right angle, $E$ and $D$ are points on opposite side of $A C$, with $E$ on the same side of $A C$ as $B$, scuh that $\triangle A C D$ and $\Delta B C E$ are both equilateral.Prove that $\alpha(\triangle B C E)=2 \alpha(\triangle A C D)$. $\mbox{ (5 marks)}$

6.$A B C$ is a triangle such that $B C: C A: A B=25: 24: 7$.If $B P C, C Q A$ and $A R B$ are equilateral triangles, prove that $\alpha(\triangle B P C)=\alpha(\Delta C Q A)+\alpha(\triangle A R B)$. $\mbox{ (5 marks)}$

7.In the figure, $P A$ is a tangent segment and $P B C$ is a secant segment.Prove that $\frac{A B^{2}}{C A^{2}}=\frac{P B}{P C}$. $\mbox{ (5 marks)}$

$\quad\;\,$$\, 1.\alpha(\Delta X Y S): \alpha(\triangle A B S)=16: 25 2.\alpha(\triangle A B S): \alpha(\Delta X Y S)=25: 16 3.\alpha(\triangle B R Q)=8 \mathrm{~cm}^{2} / \alpha(P A Q R)=10 \mathrm{~cm}^{2} 4.Prove 5.Prove 6.Prove 7.Prove ## Group (2010) \quad\;\,$$\,$
1.In $\triangle A B X, C$ is a point on the segment $B X$ and $D$ is a point on the segment $A X$ such that $\angle B A C=\angle B D C.$ Prove that $\alpha(\triangle A B X): \alpha(\Delta C D X)=A B^{2}: C D^{2}$.$\text{ (5 marks)}$

2.In $\triangle A B C, D$ is a point of $A C$ such that $A D=C D.E$ is on $B C$ such that $D E / / A B.$ Compare the areas of $\triangle A B C$ and $\triangle C D E.$ If $\alpha(A B E D)=30$, what is $\alpha(\triangle A B C) ?$ $\text{ (5 marks)}$

3.In $\triangle A B C, D$ is a point of $A C$ such that $A D=C D.E$ is on $B C$ such that $D E / / A B.$ Compare the areas of $\triangle C D E$ and $\triangle A B C.$ If $\alpha(A B E D)=30$, what is $\alpha(\triangle A B C) ?$ $\text{ (5 marks)}$

4.In a trapezium $A B C D, A B$ is twice $D C$ and $A B / / D C.$ If $A C$ and $B D$ intersect at $P$, prove that $\alpha(\triangle A P B)=4 \alpha(\Delta C P D).$ $\text{ (5 marks)}$

5.$P Q R S$ is a trapezium in which $P Q / / R S$ and $\angle P S Q=\angle R.$ Prove that $P S^{2}: Q R^{2}=P Q: R S$.$\text{ (5 marks)}$

6.In trapezium $A B C D$ the diagonals $A C$ and $B D$ intersect at $O.$ If $A B / / D C$ and $9 \alpha(\triangle A \mathrm{O} B)=16 \alpha(\triangle C \mathrm{OD})$, find the ratios $A B: C D$ and $\alpha(\triangle A O D): \alpha(\Delta C O D)$.$\text{ (5 marks)}$

7.In $\triangle P Q R, S$ and $T$ are the points on the sides $P Q$ and $P R$ respectively, and $S T \| Q R.$ If $P S=5, S Q=10$ and $\alpha(S Q R T)=104$, find $\alpha(\Delta P Q R)$.$\text{ (5 marks)}$

8.In $\triangle P Q R, P Q=6, P R=9$, and $S$ is a point on $P R$ such that $\angle P Q S=\angle R.$ Given that $\alpha(\triangle P Q S)=20$, calculate $\alpha(\Delta Q R S).$ $\text{ (5 marks)}$

9.In $\triangle A B C, A D$ and $B E$ are the altitudes.If $4 \alpha(\triangle D E C)=3 \alpha(\triangle A B C)$, find $\angle A C B.$ $\text{ (5 marks)}$

$\quad\;\,$
1.Prove
2.$\alpha(\triangle A B C): \alpha(\triangle C D E)=4: 1 ; \alpha(\triangle A B C)=40$
3.$\alpha(\triangle C D E): \alpha(\triangle A B C)=1: 4 ; \alpha(\triangle A B C)=40$
4.Prove
5.$A B: C D=4: 3 ; \alpha(\triangle AOD): \alpha(\triangle COD)=4: 3$
6.Prove
7.$117$
8.$25$
9.$30^{\circ}$