SAS Similarity

Q. Question 14
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 $\Delta ABC\sim \Delta PQR$.

Q. Question 1
If two lines intersect, prove that the vertically opposite angles are equal.

Q. In the given figure ABC is a right-angled triangle with $\angle BAC={90}^{\circ }$ [4 MARKS]

i) Prove that $\Delta ADB\sim \Delta CDA$
ii) IF BD = 18 cm, CD = 8 cm, find AD
iii) Find the ratio of the area of $\Delta ADB$ is to area of $\Delta CDA$

Q.

Question 13
If an isosceles $\Delta$ ABC in which AB = AC = 6cm, is inscribed in a circle of radius 9 cm, find the area of the triangle.

Q.

The angle between the two lines $y-2x=9\text{and}x+2y=7,$ is

1. $60°$

2. $30°$

3. $90°$

4. $45°$

Q. Question 12
Sides AB and BC and median AD of a triangle ABC are respectively proportional to sides PQ and QR and median PM of triangle PQR. Show that $\Delta ABC\sim \Delta PQR$.

Q.

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

Q.

CM and RN are respectively the medians of $\Delta ABC$ and $\Delta PQR$. If $\Delta ABC\sim \Delta PQR$, prove that:

(i) $\Delta AMC\sim \Delta PNR$

(ii) $\frac{CM}{RN}=\frac{AB}{PQ}$

(iii) $\Delta CMB\sim \Delta RNQ$
[3 MARKS]

Q.

If $AD$ and $PM$ are medians of triangles $ABC$ and $PQR$, respectively where $\Delta ABC~\Delta PQR$ prove that $\frac{AB}{PQ}=\frac{AD}{PM}$.

Q.

In a triangle, the lengths of two larger sides are $10cm$and $9cm$. If the angles of the triangle are in AP, then the length of the third side is:

1. $\sqrt{5}-\sqrt{6}$

2. $\sqrt{5}+\sqrt{6}$

3. $\sqrt{5}\pm \sqrt{6}$

4. $5\pm \sqrt{6}$

Q.

It is given that $\Delta DEF~\Delta RPQ$. Is it true to say that $\angle D=\angle R$ and $\angle F=\angle P$? Why?

1. True
2. False

Q. Question 5
In figure, two line segments AC and BD intersect each other at the point P such that $PA=6cm,PB=6cm,PC=3cm,PC=2.5cm,PD=5cm,\angle APB={50}^{\circ }and\angle CDP={30}^{\circ }.Then\angle PBA$ is equal to




(A) ${50}^{\circ }$
(B) ${30}^{\circ }$
(C) ${60}^{\circ }$
(D) ${100}^{\circ }$

Q.

In an isosceles $\Delta ABC$, the base AB has produced both ways in P and Q such that $AP\times BQ=A{C}^2$.

Prove that $\Delta ACP\sim \Delta BCQ$.

Q.

Calculate the percentage of p-character in the orbital occupied by the lone pairs in water molecules : [Given:∠HOH is 104.5∘ and cos (104.5)= − 0.25]

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 12
Is it true to say that if in two triangles, an angle of one triangle is equal to an angle of another triangle and two sides of one triangle are proportional to the two sides of the other triangle, then the triangles are similar? Give reasons for your answer.

Q. ABC is a right-angled triangle with $\angle ABC={90}^{\circ }$. D is any point on AB and DE is perpendicular to AC. Prove that: [4 MARKS]
i) $\Delta ADE\sim \Delta ACB$
ii) If AC = 13 cm, BC = 5 cm and AE = 4 cm, find DE and AD.
iii) Find, area of $\Delta ADE$ : area of quadrilateral BCED

Q. Question 16
If AD and PM are medians of triangles ABC and PQR, respectively where $\Delta ABC\sim \Delta PQR$. Prove that $\frac{AB}{PQ}=\frac{AD}{PM}$.

Q. Question 5
To construct a triangle similar to a given $\Delta$ABC with its sides $\frac{8}{5}$ of the corresponding sides of $\Delta$ABC draw a ray BX such that $\angle$ CBX is an acute angle and X is on the opposite side of A with respect to BC. The minimum number of points to be located at equal distances on ray BX is
(A) 5
(B) 8
(C) 13
(D) 3

Q.

In the given figure, $\angle 1=\angle 2$ and $\frac{AC}{BD}=\frac{CB}{CE}$.

Prove that $\Delta ACB\sim \Delta DCE$.

Q.

Question 4
In the following figure, DE||AC and DF||AE. Prove that $\frac{BF}{FE}=\frac{BE}{EC}$ .

Q.

State the SAS-similarity criterion.

Q.

In triangle ABC, points D and E lie on sides AB and AC respectively such that DE is parallel to BC.

If AB = 3.6 cm, AC = 2.4 cm and AD = 2.1 cm, then AE is equal to ____.

1. $1.4cm$

2. $1.05cm$

3. $1.8cm$

4. $1.2cm$

Q. In the given figure, $\angle QPS=\angle RPTand\angle PRQ=\angle PTS$. [4 MARKS]
(i) Prove that $\Delta PQR\sim \Delta PST$
(ii) If PT:ST=3:4, find QR : PR.

Q. A horse is tied to a post by a rope. If the horse moves along a circular path always keeping the rope tight and describes $88metres$ when it has traced out ${72}^o$ at the center, find the length of the rope.

Q. The two triangles are congruent by _________ congruency rule.

1. SSS
2. SAS
3. RHS
4. ASA

Q.

Triangle ABC is similar to triangle PQR. If AD and PM are corresponding medians of the two triangles, prove that : $\frac{AB}{PQ}=\frac{AD}{PM}$.

Q. In the given figure, $\angle GHE=\angle DFE={90}^{\circ },DH=8cm,DF=12cm,DG=(3x-1)cm$ and $DE=(4x+2)cm$. find the length of segments DG and DE.
[4 MARKS]

Q. Question 3
In the following figure, if LM || CB and LN || CD, prove that $\frac{AM}{AB}=\frac{AN}{AD}$.

Q. Question 15
O is the point of intersection of the diagonals AC and BD of a trapezium ABCD with ABIIDC. Through O, a line segment PQ is drawn parallel to AB meeting AD in P and BC in Q. Prove that PO = QO.