Tests for Similarity of Triangles

Q.

In triangle ABC, DE is parallel to BC; where D and E are the points on AB and AC respectively. Also, if AD = 12 cm, BD = 24 cm and BC = 8 cm, the length of DE is

1. $\frac{3}{4}\text{cm}$

2. $\frac{2}{3}\text{cm}$

3. $2\frac{2}{3}\text{cm}$

4. $2\frac{3}{4}\text{cm}$

Q. Question 12
If S is a point on the side PQ of a $\Delta$ PQR such that PS = QS = RS, then

$\left(A\right)\left(PR\right)\left(QR\right)=R{S}^2$
$\left(B\right)Q{S}^2+R{S}^2=Q{R}^2$
$\left(C\right)P{R}^2+Q{R}^2=P{Q}^2$
$\left(D\right)P{S}^2+R{S}^2=P{R}^2$

Q.

Two poles of height a meters and b meters are p meters 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 $\frac{ab}{a+b}$ meters.

Q. A vertical stick 10 cm long casts a shadow 8 cm long. At the same time, a tower casts a shadow 30 m long. Determine the height of the tower.

1. 37.5 cm
2. 39.7 cm
3. 30 cm
4. 35.4 cm

Q. In a $△ABC$, points P and Q are on sides AB and AC respectively. If AP = 3 cm, PB = 6 cm. AQ = 5 cm and QC = 10 cm, then BC = ____.

1. 4PQ
2. $\frac{PQ}{2}$
3. 3PQ
4. $P{Q}^2$

Q.

$\text{If in}△CAB\text{and}△FED,$

$\frac{AB}{EF}=\frac{BC}{FD}=\frac{AC}{ED}$,

$\text{then which of the following is true?}$

1. $△CAB\sim △EFD$

2. $△CAB\sim △FED$

3. $△ABC\sim △EFD$

4. $△ABC\sim △DEF$

Q.

$P$ is a point on the segment joining the feet of two vertical poles of heights $a$ and $b$. The angles of elevation of the tops of the poles from $P$ are ${45}^o$ each. Then, the square of the distance between the tops of the poles is

1. $\frac{(a^2+b^2)}{2}$

2. $\left(a^2+b^2\right)$

3. $2(a^2+b^2)$

4. $4(a^2+b^2)$

Q.

Angles BAC of triangle ABC is obtuse and AB = AC. P is a point in BC such that PC = 12 cm. PQ and PR are perpendiculars to sides AB and AC respectively.

If PQ = 15 cm and PR = 9 cm; then the length of PB is ___ cm.

Q.

A vertical stick $1m$ long casts a shadow $80cm$ long. At the same time, a tower casts a shadow $30m$ long. Determine the height of the tower.

Q. The triangles are congruent by which congruency rule?

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

Q.

In a right triangle $\Delta$ ABC, right angled at B, BD is a perpendicular dropped onto the hypotenuse AC.

If AC = 2AB, find the area of $\Delta$ ABD, given, area of $\Delta$ ABC = 5 sq units.

1. 0.5 sq. units

2. 1 sq. unit

3. 1.25 sq. units

4. 5 sq. units

Q.

A vertical stick 24 cm long casts a shadow 16 cm long on the ground. At the same time a tower casts a shadow 80 m long on the ground. Determine the height of the tower.

1. 120m

2. 80m

3. 70m

4. 60m

Q. Are the given triangles similar? If yes, by which test?

1. SSS
2. SES
3. AAA
4. Not similar

Q. Which of the following is correct for the following triangles?

1. ∆ABC $\sim$ ∆PQR
2. ∆ACB $\sim$ ∆RQP
3. ∆ABC $\sim$ ∆RQP
4. ∆ACB $\sim$ ∆PQR

Q.

If $\Delta ABC\sim \Delta PQR$ and AD and PM are corresponding medians of the two triangles, then

1. $\frac{AB}{PM}=\frac{AD}{PQ}$

2. $\frac{AB}{PR}=\frac{AD}{PM}$

3. $\frac{AB}{PQ}=\frac{AD}{PM}$

4. $\frac{AC}{PQ}=\frac{AD}{PM}$

Q. A kid is standing 4 m away from a pole of height 7m and another pole is 8 m behind it. If the kid’s vision is 1 m from the ground, find the height of the second pole. (Assuming both poles are perpendicular to the ground)

1. 18
2. 20
3. 21
4. 19

Q.

A vertical stick 20 m long casts a shadow 10 m long on the ground. At the same time, a tower casts a shadow 50 m long on the ground. Find the height of the tower.

1. 100 m

2. 120 m

3. 25 m

4. 200 m

Q.

The minimum number of parameters required to prove similarity between two triangles is ____.

1. 1

2. 2

3. 3

4. 4

Q. Given : $RS$ and $PT$ are altitudes of $\Delta PQR$. Prove that :

ii) $PQ\times QS=RQ\times QT$

Q. In a right angle triangle $\Delta$ABC, a perpendicular BD is drawn on to the largest side from the opposite vertex. Which of the following does NOT give the ratio of the areas of $\Delta$ABD and $\Delta$ BCD?

1. ${\left(\frac{AB}{BC}\right)}^2$
2. ${\left(\frac{AB}{AC}\right)}^2$
3. ${\left(\frac{AD}{BD}\right)}^2$
4. ${\left(\frac{BD}{DC}\right)}^2$

Q.

In the figure given below, if BD = 2.4 cm, AC = 3.6 cm, PD = 4 cm, PB = 3.2 cm, and AC is parallel to BD, then the lengths of PA & PC will be

1. 4.8 cm and 6 cm respectively

2. 6.8 cm and 4 cm respectively

3. 6 cm and 4.8 cm respectively

4. 4 cm and 6.8 cm respectively

Q.

In the figure given below AB, CD, and EF are parallel lines. Given AB = 9 cm, DC = y cm, EF = 4.5 cm, BC = x cm, and CE = 3 cm. The value of 'x' and 'y' are:

1. 6 cm and 5 cm respectively

2. 4.5 cm and 3 cm respectively

3. 3 cm and 6 cm respectively

4. 6 cm and 3 cm respectively

Q.

In the figure given below AB, CD, and EF are parallel lines. Given AB = 9 cm, DC = y cm, EF = 4.5 cm, BC = x cm, and CE = 3 cm. The value of 'x' and 'y' are:

1. 3 cm and 6 cm respectively

2. 6 cm and 5 cm respectively

3. 4.5 cm and 3 cm respectively

4. 6 cm and 3 cm respectively

Q.

Sides of two similar triangles are in the ratio of 4: 9. Areas of these triangles are in the ratio?

1. 4:9

2. 2:3

3. 81:16

4. 16:81

Q. A vertical stick 10 cm long casts a shadow 8 cm long. At the same time, a tower casts a shadow 30 m long. Determine the height of the tower.

Q. In a right triangle $\Delta$ ABC with right angle at B, BD is a perpendicular, dropped onto the hypotenuse. Which of the following statements are true?

1. $\Delta ADB$ ~ $\Delta ABC$
2. $\Delta ADB$ ~ $\Delta BDC$
3. $\Delta ABC$ ~ $\Delta BDC$
4. $\text{None of these}$

Q. The angle of elevation of a tower from a point is ${30}^0.$ At a point on the horizontal line passing through the foot of 'the tower and 50 metros nearer it, the angle of elevation is ${60}^0.$ The distance of the first point from the tower is

1. 75 metres
2. 50 metres
3. 100 metres
4. 150 metres

Q. State whether the following pair of triangle is similar or not. Write the similarity criterion used and write it in symbolic form.

Q. Given : $RS$ and $PT$ are altitudes of $\Delta PQR$. Prove that :

i) $\Delta PQT$ ∼ $\Delta RQS$

Q. The angle of elevation of the top of a cable tower when
observed from the top of a 9 m tall building is 60° and the angle of depression of the foot of this tower is 45°. Determine the height of the cable tower.​​