Construction of a Similar Triangle, When One Vertex Is Common

Q. Draw a triangle ABC with BC = 7 cm, ∠B = 45° and ∠C = 60°. Then construct the another triangle, whose sides are $\frac{3}{5}$ times the corresponding sides of △ABC.

Q. Draw a triangle ABC in which AB = 5 cm, BC = 6 cm and $\angle$ ABC = ${60}^{\circ }$. Then, construct a triangle whose sides are $\frac{5}{7}$ times the corresponding sides of $\Delta$ ABC. Justify your construction. [4 MARKS]

Q. A triangle similar to given $\Delta ABC$with sides equal to $\frac{3}{4}$ of the sides of $\Delta ABC$ is to be constructed as the given image. Arrange the following steps of construction in order:

1. Join $B_{4}C$ and raw a line through $B_3$( the third point, 3 being smaller of 4 in$\frac{3}{4}$) parallel to $B_{4}C$ to intersect BC at C'.

2. Draw any ray BX making an acute angle with BC on the side opposite to the vertex A.

3. Locate 4(the greater of 3 and 4 in$\frac{3}{4}$) points $B_1,B_2,B_3\text{and}{B}_4$ on BX so that $B{B}_1=B_1{B}_2=B_2{B}_3=B_3{B}_4$

4. Draw a line through C' parallel to the line CA to intersect BA at A'.

1. 1, 2, 3, 4
2. 4, 3, 2, 1
3. 2, 3, 1, 4
4. 2, 1, 3, 4

Q.

The ratio of $\frac{BC}{C{C}^{\prime }}$ is 3:5. What will the ratio of corresponding sides be if triangles ABC and A'BC' are similar? Given that BC and BC' lie
on the same ray BY.

1. 3:8

2. 5:8

3. 8:3

4. 8:5

Q. Question 7 (ii)
In the below figure, altitudes AD and CE of $\Delta ABC$ intersect each other at the point P. Show that

(ii) $\Delta ABD\sim \Delta CBE$

Q.

$\Delta ABC$ of dimesions $AB=4cm,BC=5cm$ and ∠B= 60^o is given.
A ray $BX$ is drawn from $B$ making an acute angle with $AB$.
5 points $B_1,B_2,B_3,B_4\&B_5$ are located on the ray such that $BB1=B1B2=B2B3=B3B4=B4B5$.
$B_4$ is joined to $A$ and a line parallel to $B_{4}A$ is drawn through $B_5$ to intersect the extended line $AB$ at $A^{\prime }$.
Another line is drawn through $A^{\prime }$ parallel to $AC$, intersecting the extended line $BC$ at $C^{\prime }$.
Find the ratio of the corresponding sides of $\Delta ABC$ and $\Delta A^{\prime }B{C}^{\prime }$.

1. $4:1$

2. $4:5$

3. $1:5$

4. $1:4$

Q.

Match the following based on the construction of similar triangles, if scale factor is $\left(\frac{m}{n}\right)$.

$\begin{array}{cc}I.\frac{m}{n}>1& a)Thesimilartriangleissmallerthantheoriginaltriangle.\\ II.\frac{m}{n}<1& b)Thetwotrianglesarecongruenttriangles.\\ III.\frac{m}{n}=1& c)Thesimilartriangleislargerthantheoriginaltriangle.\end{array}$

1. I- a, II - b, III - c

2. I- c, II - a, III - b

3. I- a, II - c, III - b

4. I- b, II - a, III - c

Q. Draw a triangle ABC in which AB = 4cm , angle B = 60 and altitude CL = 3cm. Construct a triangle ADE similar to triangle ABC such that each side of triangle ADE is $\frac{3}{2}$ times that of the corresponding side of triangle ABC

Q. Question 4
In the figure,
$\frac{QR}{QS}=\frac{QT}{PR}and$
$\angle 1=\angle 2$.
$Showthat\Delta PQS\sim \Delta TQR$.

Q. Question 1
In figure, if $\angle BAC={90}^{\circ }andAD\perp BC$. Then,


(A) $BD.CD=B{C}^2$
(B) $AB.AC=B{C}^2$
(C) $BD.CD=A{D}^2$
(D) $AB.AC=A{D}^2$

Q. Construction of a $\Delta A^{\prime }{C}^{\prime }B$ similar to $\Delta ACB$ is shown in the figure below, which condition is used to construct $A^{\prime }{C}^{\prime }$ ||$AC$?

1. Corresponding angles are equal
2. Alternate interior angles are equal
3. Co-interior angles are supplementary
4. Perpendicular bisector theorem

Q. To construct a triangle similar to a given triangle with a scale factor $\frac{3}{5}$, 8 points are to be marked on a ray drawn at an acute angle to one of the sides.

1. TRUE
2. FALSE

Q.

The scale factor for constructing similar triangles is given by the __ of corresponding sides of triangles.

Q.

Draw a triangle ABC with side BC = 7 cm, ∠B = 45°, ∠A = 105°. Then, construct a triangle whose sides are times the corresponding side of ΔABC. Give the justification of the construction.

Q. Construct a triangle ABC with side BC = 7 cm , $\angle B={45}^{\circ },\angle A={105}^{\circ }$. Then cnstruct another triangle whose sides are $\frac{3}{4}$ times the corresponding sides of the $\Delta ABC$

Q. A triangle $ABC$ with $BC=6cm,AB=5cm$ and $\angle ABC={60}^{\circ }$. The image of constructing a similar triangle of $\Delta ABC$ whose sides are $\frac{3}{4}$ times the corresponding sides of the $\Delta ABC$ is given below.


Arrange the steps of construction in the correct order.
1. Join $B_{4}C$ and raw a line through $B_3$
(the third point, 3 being smaller of 4 in $\frac{3}{4}$) parallel to $B_{4}C$ to intersect BC at C'.

2. Draw any ray $BX$ making an acute angle with BC on the side opposite to the vertex A.

3. Locate 4 (the greater of 3 and 4 in$\frac{3}{4}$) points $B_1,B_2,B_3\text{and}{B}_4$ on BX so that $B{B}_1=B_1{B}_2=B_2{B}_3=B_3{B}_4$

4. Draw a line through C' parallel to the line CA to intersect BA at A'.

1. 1, 2, 3, 4
2. 2, 1, 3, 4
3. 2, 3, 1, 4
4. 2, 4, 1, 3

Q.

A triangle ABC has base angle ${45}^{\circ }$. The perimeter is [2+$\sqrt{2}$]. What type of triangle is this?

1. Isoceles triangle

2. Right triangle

3. Scalene triangle

4. Can't be determined

Q. If you are asked to construct $△$APQ $\sim$ $△$ABC with the scale factor $\frac{3}{5}$, which of the following are correct? In $△ABC,AB=4cm$, $BC=3cm$ and $\angle ABC={90}^{\circ }$.​

1. $\angle AQP=\angle ACB$
2. $\angle A{A}_{3}P=\angle A{A}_{5}P$
3. $AP:PB=3:2$
4. 5PQ = 3BC

Q. A triangle ABC with sides BC =7cm, $\angle B={45}^{\circ },\angle A={105}^{\circ }$ is given. The image of constructing a similar triangle of $\Delta ABC$ whose sides are $\frac{3}{4}$ times the corresponding sides of the $\Delta ABC$ is given below. Then $\frac{A^{\prime }{C}^{\prime }}{AC}$ is equal to :

1. $\frac{3}{4}$
2. $\frac{4}{3}$
3. $\frac{3}{7}$
4. $\frac{4}{7}$

Q. Question 6
Draw a triangle ABC with side $BC=7cm,\angle B={45}^{\circ },\angle A={105}^{\circ }$. Then, construct a triangle whose sides are 4/3 times the corresponding sides of $\Delta ABC$.

Q. A $\Delta ABC$ with AB = 6 cm, $\angle A={30}^{\circ }$ and $\angle B={60}^{\circ }$ is given. Another $\Delta A{B}^{\prime }{C}^{\prime }$ similar to $\Delta ABC$ with AB' = 8 cm is constructed as shown below:

The reason why we don't construct a ray making an acute angle at A is

1. Scale is not given.
2. $\Delta ABC$ is a right-angled triangle.
3. Length of AB' is given.
4. $\angle A={30}^{\circ }$

Q. In $\Delta ABC,AB=BC=6cm$. A circle is drawn with center at B and radius 2 cm. What is the scale factor of the triangle formed by the point B and the points where the circle intersects $△ABC$?

1. $\frac{2}{3}$
2. $1$
3. $\frac{1}{3}$
4. $\frac{1}{2}$

Q. $\Delta PQR\sim \Delta LTR$. In $\Delta PQR$, PQ = 4.2 cm, QR = 5.4 cm, PR = 4.8 cm.
Construct $\Delta PQR$ and $\Delta LTR$, such that $\frac{RQ}{RT}=\frac{7}{5}$, then the measure of LT is equal to ____ cm.

1. 5.2 cm
2. 4.8 cm
3. 3 cm
4. 3.6 cm

Q.
The postulate used in the division of line segment to draw $A_{3}C$ parallel to $A_{5}B$ is :

1. Corresponding angles are equal
2. Alternate interior angles are equal
3. Co-interior angles are supplementary
4. Perpendicular bisector theorem

Q. Draw a triangle ABC with side BC = 7 cm, $\angle$ B = ${45}^{\circ }$, and $\angle$ A = ${105}^{\circ }$. Then, construct a triangle whose sides are $\frac{4}{3}$ times the corresponding sides of $\Delta$ ABC [3 MARKS]

Q. Draw a triangle ABC in which AB = 5 cm, BC = 6 cm and $\angle$ ABC = ${60}^{\circ }$. Then, construct a triangle whose sides are $\frac{5}{7}$ times the corresponding sides of $\Delta$ ABC. Justify your construction. [4 MARKS]

Q.

Construct a triangle ABC in which BC is 5 cm, Angle B is ${60}^{\circ }$ and AB+AC=10cm. What type of triangle is this?

Q.

Given that

Triangle 1: 3 cm, 3 cm, 4 cm

Triangle 2: 6 cm, 6 cm, 8 cm

Triangle 3: 12 cm, 12 cm, 16 cm

Triangle 4: 24 cm, 24 cm, 32 cm

If Triangle 1 is the original triangle, what scale factor is used to generate Triangle 2, Triangle 3, and Triangle 4 from Triangle 1?

1. Triangle 2: scale factor = 2
   Triangle 3: scale factor = 4
   Triangle 4: scale factor = 6

2. Triangle 2: scale factor = 2
   Triangle 3: scale factor = 4
   Triangle 4: scale factor = 8

3. Triangle 2: scale factor = 2
   Triangle 3: scale factor = 3
   Triangle 4: scale factor = 6

4. Triangle 2: scale factor = 3
   Triangle 3: scale factor = 2
   Triangle 4: scale factor = 8

Q.

Given are the steps to construct tangents from an external point P to a circle with centre O. Order the steps correctly.
STEPS:
i) Join PM and PN.

ii) Draw a circle with centre as L and radius LO. Let it cut the given circle at points M and N.

iii) Bisect the segment OP, at L.

iv) Join OP.

Q.

If you are asked to construct APQ ~ ABC in the scale factor $\frac{3}{5}$. In $△ABC$, AB = 4 cm, BC = 3 cm and $\angle ABC$ = ${90}^{\circ }$.