The famous mathematician who gave an important truth called "Basic proportionality theorem" belongs to:
- AChina
- BIndia
- CBabylonia
- ✓Greece
Answer: D.
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Triangles question types
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M.C.Q (1 Marks)
34 Q→021 Marks Question
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20 Q→055 Marks Questions
11 Q→06Case study (4 Marks)
3 Q→Sample Questions
Triangles questions
One sample from each question group in this chapter. Select any group above to see the full set with answer keys.
$\triangle D E F \sim \triangle A B C$; If $D E: A B=2: 3$ and $\operatorname{ar}(\triangle D E F)$ is equal to $44 $ square units, then area $(\triangle A B C)$ in square units is
- ✓$99$
- B$120$
- C$\frac{176}{9}$
- D$66$
Answer: A.
View full solution →In a $\triangle ABC , \angle A =x^{\circ}, \angle B =(3 x-2)^{\circ}, \angle C =y^{\circ}$. Also $\angle C -\angle B =9^{\circ}$. The sum of the greatest and the smallest angles of this triangle is
- ✓$107^{\circ}$
- B$135^{\circ}$
- C$155^{\circ}$
- D$145^{\circ}$
Answer: A.
View full solution →In $\triangle ABC$ and $\triangle DEF , \angle F =\angle C , \angle B =\angle E$ and $AB =\frac{1}{2} DE$. Then the two triangles are
- ACongruent, but not similar
- ✓Similar, but not congruent
- CNeither congruent nor similar
- DCongruent as well as similar
Answer: B.
View full solution →In figure, $DE \| BC , AD =2 cm$ and $BD =$ 3 cm , then $\operatorname{ar}(\triangle ABC ): \operatorname{ar}(\triangle ADE )$ is equal to
- A$4: 25$
- B$2: 3$
- C$9: 4$
- ✓$25: 4$
Answer: D.
View full solution →In a $\triangle A B C, D E \| B C$, then find the value of $x$.
View full solution →In $\triangle A B C, D$ and $E$ are point on side $A B$ and $A C$ resp. such that $DE \| BC$. If $A E=2 \ cm, A D=3 \ cm$ and $B D=4.5 \ cm$ then find $C E$.
View full solution →In a, right angled at $B, A C-A B=2 \ cm, B C=8 \ cm$ , find the value of $A C$.
View full solution →Find the length of the diagonal of a square whose each side is $8 \ cm .$
View full solution →In below fig., $MN \| BC$ and $AM : MB =1: 2$, then $\frac{\operatorname{ar}(\triangle AMN )}{\operatorname{ar}(\triangle ABC )}=.......$
View full solution →In Figure $4,$ a triangle $A B C$ is drawn to circumscribe a circle of radius $3 \ cm$ , such that the segments $B D$ and $D C$ are respectively of lengths $6 \ cm$ and $9 \ cm$ . If the area of $\triangle A B C$ is $54$ , then find the lengths of sides $A B$ and $A C$.
View full solution →State which of the two triangles given in the figure are similar. Also state the similarity criterion used.
In the figure, $\triangle O A D \sim \triangle O B C$, If $\angle A O C=90^{\circ}$ and $\angle O B C=30^{\circ}$, find $\angle O D A$ and $\angle C O B$
View full solution →In $\triangle A B C, \angle A=90^{\circ} A N \perp B C, B C=13 \ cm$ and $A C=5 Cm$. Find the ratio of $\operatorname{ar}(\triangle N A C): \operatorname{ar}(\triangle A B C)$
View full solution →In the given figure, if $A B \| DC$, find the value of $x$.
View full solution →State and prove Py thagoras theorem.
In the given figure $\angle B=\angle D=90^{\circ}$. If $A B=12 \ cm , A C=13 \ cm, C E=5 \ cm$ and $E D=4 \ cm,$ find the length of $B D$.
View full solution →In the given figure $\triangle A B D \sim \triangle P Q S$ when $A D$ and $P S$ are medians. Prove that $\triangle A B C \sim \triangle P Q R$.
View full solution →In the figure if $C D=17 m, B D=8 m$ and $A D=4 m, $ then find the value of $A C$.
View full solution →In the figure $A B|\mid Q R$ and $B C| \mid R S$. Prove that
$\frac{P A}{P Q}=\frac{P C}{P S}$
View full solution →$\frac{P A}{P Q}=\frac{P C}{P S}$
In $\triangle A B C$, If $\angle A D E=\angle B$, then prove that $\triangle A D E \sim \triangle A B C$.
Also if $A D=7.6 cm, A E=7.2 cm, B E=4.2 cm$ and $B C=5.4 cm$, then find $D E$.
View full solution →Also if $A D=7.6 cm, A E=7.2 cm, B E=4.2 cm$ and $B C=5.4 cm$, then find $D E$.
In the given figure $\triangle A B C$ is an equilateral. D is a point on $B C$ such that $B D=\frac{1}{3} B C$ Prove that $9 A D^2=7 A B^2$
View full solution →In the given figure, $A B C$ and $D B C$ are two triangles on the same base $B C$. If $A D$ intersects $B C$ at $O$, show that $\frac{\operatorname{ar}(\triangle A B C)}{\operatorname{ar}(\triangle D B C)}=\frac{A O}{D O}$
View full solution →Two poles of height $p$ and $q$ metres are standing vertically on a level ground, a metres apart. Prove that the height of the point of intersection of the lines joining the top of each pole to the foot of the opposite pole is given by $\frac{p q}{p+q}$ metres.
View full solution →
Quilts are available in various colours and design. Geometric design includes shapes like squares, triangles, rectangles, hexagons etc.
One such design is shown above. Two triangles are highlighted, $\triangle ABC$ and $\triangle PQR .$
Based on above information, answer the following questions:
$1.$ Which of the following criteria is not suitable for $\triangle ABC$ to be similar to $\triangle QRP ?$
$(a) \ \text{SAS} \ (b) \ \text{AAA}$
$(c) \ \text{SSS} \ (d) \ \text{RHS}$
$2.$ If each square is of length $x$ unit, then length $B C$ is equal to
$(a) \ x \sqrt{2}$ unit $(b) \ 2 x$ unit
$(c) \ 2 \sqrt{x}$ unit $(d)\ x \sqrt{x}$ unit
$3.$ Ratio $BC : PR$ is equal to
$(a) \ 2: 1 \ (b)\ 1: 4 $
$(c) \ 1: 2 \ (d)\ 4: 1$
$4. \operatorname{ar}( PQR ): \operatorname{ar}( ABC )$ is equal to
$(a) \ 2: 1 \ (b) \ 1: 4 $
$(c) \ 4: 1 \ (d) \ 1: 8$
$5.$ Which of the following is not true?
$(a) \ \triangle TQS \sim \triangle PQR \ (b) \ \triangle CBA \sim \Delta STQ $
$(c) \ \triangle BAC \sim \triangle PQR \ (d) \ \triangle PQR \sim \triangle ABC$
A farmer has a field in the shape of trapezium, whose map with scale 1 cm = 20 m, is given below:
The field is divided into four parts by joining the opposite vertices.
Based on the above information, answer any four of the following questions:
1. The two triangular regions AOB and COD are
(a) Similar by AA criterion (b) Similar by SAS criterion
(c) Similar by RHS criterion (d) Not similar
2. The ratio of the area of the $\triangle A O B$ to be area of $\triangle COD$, is
(a) $4: 1$ (b) $1: 4$
(c) $1: 2$ (d) $2: 1$
3. If the ratio of the perimeter of $\triangle A O B$ to the perimeter of $\triangle C O D$ would have been $1: 4$, then
(a) $AB =2 CD$ (b) $AB =4 CD$
(c) $CD =2 AB$ (d) $CD =4 AB$
4. If in $\triangle s A O D$ and $B O C, \frac{A O}{B C}=\frac{A D}{B O}=\frac{O D}{O C}$, then
(a) $\triangle AOD \sim \triangle BOC$ (b) $\triangle AOD \sim \triangle BCO$
(c) $\triangle ADO \sim \triangle BCO$ (d) $\triangle ODA \sim \triangle OBC$
5. If the ratio of areas of two similar triangles $A O B$ and COD is $1: 4$, then which of the following statements is true?
(a) The ratio of their perimeters is $3: 4$ (b) The corresponding altitudes have a ratio $1: 2$
(c) The medians have a ratio $1: 4$ (d) The angle bisectors have a ratio $1: 16$
View full solution →The field is divided into four parts by joining the opposite vertices.
Based on the above information, answer any four of the following questions:
1. The two triangular regions AOB and COD are
(a) Similar by AA criterion (b) Similar by SAS criterion
(c) Similar by RHS criterion (d) Not similar
2. The ratio of the area of the $\triangle A O B$ to be area of $\triangle COD$, is
(a) $4: 1$ (b) $1: 4$
(c) $1: 2$ (d) $2: 1$
3. If the ratio of the perimeter of $\triangle A O B$ to the perimeter of $\triangle C O D$ would have been $1: 4$, then
(a) $AB =2 CD$ (b) $AB =4 CD$
(c) $CD =2 AB$ (d) $CD =4 AB$
4. If in $\triangle s A O D$ and $B O C, \frac{A O}{B C}=\frac{A D}{B O}=\frac{O D}{O C}$, then
(a) $\triangle AOD \sim \triangle BOC$ (b) $\triangle AOD \sim \triangle BCO$
(c) $\triangle ADO \sim \triangle BCO$ (d) $\triangle ODA \sim \triangle OBC$
5. If the ratio of areas of two similar triangles $A O B$ and COD is $1: 4$, then which of the following statements is true?
(a) The ratio of their perimeters is $3: 4$ (b) The corresponding altitudes have a ratio $1: 2$
(c) The medians have a ratio $1: 4$ (d) The angle bisectors have a ratio $1: 16$
Observe the figures given below carefully and answer the questions:
(i) Name the figure(s) wherein two figures are similar.
(ii) Name the figure(s) wherein the figures are congruent.
(iii)(a) Prove that congruent triangles are also similar but not the converse.
OR
(b) What more is least needed for two similar triangles to be congruent?
(i) Name the figure(s) wherein two figures are similar.
(ii) Name the figure(s) wherein the figures are congruent.
(iii)(a) Prove that congruent triangles are also similar but not the converse.
OR
(b) What more is least needed for two similar triangles to be congruent?
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