AA Similarity Criterion

Q.

$△$ABC is a right angled triangle, right angled at B. BD is a perpendicular to AC as shown in the figure. If AD:DC = 1:2, then $\frac{{AB}^2}{{BC}^2}$ is ____.

1. 1:4

2. 2:1

3. 4:1

4. 1:2

Q.

$\text{In the given figure,}AX:XB=3:5\text{and}XY||BC.$

$\text{If XY = 18 cm, then BC = \_\_\_ .}$

1. $36cm$

2. $40cm$

3. $44cm$

4. $48cm$

Q. In the given figure, ABCD is a quadrilateral such that $DA\perp AB$ and $CB\perp AB$, x = y and EF || AD such that EF bisects $\angle DEC$. Then $C{D}^2$ is equal to

1. $2C{E}^2$
2. $2D{E}^2$
3. Both (a) and (b)
4. None of these

Q. $\Delta ABC\sim \Delta PQR$. If $\angle B$ of $\Delta ABC$ and $\angle R$ of $\Delta PQR$ are ${60}^{\circ }$, then $\angle$ RPQ =

1. ${120}^{\circ }$
2. ${60}^{\circ }$
3. ${30}^{\circ }$

Q.

If AD, BE and CF are the medians of an equilateral triangle ABC, then which of the following statements is true?

1. 2 (AB² + BC² + AC²) = 3 (AD² + BE² + CF²)

2. AB² + BC² + AC² = AD² + BE² + CF²

3. 3 (AB² + BC² + AC²) = 4 (AD² + BE² + CF²)

4. AB² + BC² + AC² = 3 (AD² + BE² + CF²)

Q. A boy of height 60 cm is walking away from the base of a building at a speed of 2.2 m/s. If the height of the building is 7.2 m, then find the length of his shadow after 5 seconds.

1. $2m$
2. $1m$
3. $4m$
4. $3m$

Q. $\Delta$ ADE $\sim$ $\Delta$ ABC by AA Similarity criteria

1. False
2. True

Q. Minimum number of pairs of equal corresponding angles to make two triangles similar = .

1. 2
2. 3
3. 1

Q. A lamp post is situated at the middle point $M$ of the side $AC$ of a triangular plot $ABC$ with $BC=7m,CA=8cm$ and $AB=9m$. Lamp post subtends angle ${15}^o$ at the point $B$. Determine the height of the lamp post.

Q. Prove that $△ABC\sim △EDC$

Q. If $a^2=b^3=c^5=d^6$, show that, ${\log }_{d}abc=\frac{31}{5}$

Q.

In the given triangle DE II BC, which of the following statements is true?

1. $\frac{AD}{AB}=\frac{DE}{BC}$

2. $\frac{AB}{AB}=\frac{DE}{BC}$

3. $\frac{AD}{AB}=\frac{DB}{BC}$

4. $\frac{AB}{AB}=\frac{DC}{BC}$

Q. In the given figures which of the triangles are similar?

1. $\Delta FED\sim \Delta PQS$
2. $\Delta EDF\sim \Delta QPS$
3. $\Delta PQS\sim \Delta QRS$
4. $\Delta DEF\sim \Delta SQR$

Q.

In a right $△$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 $△$ABD and $△$ACB?

1. $({\frac{AB}{AC})}^2$

2. $({\frac{AD}{AB})}^2$

3. $({\frac{BD}{CB})}^2$

4. $({\frac{AB}{AD})}^2$

Q. Find the area of a triangle whose vertices are $(0,6\sqrt{3}),(\sqrt{35},7)$, and $(0,3)$.

1. $17.75$
2. $21.87$
3. $15.37$
4. $39.61$
5. $25.61$

Q. Find the value of $a$ if area of the triangle is $17$ square units whose vertices are $(0,0),(4,a),(6,4)$.

1. $a=-2$
   
2. $a=-3$
   
3. $a=3$
   
4. none of the above

Q. Use the given figure to find:
$\sin {\theta }^o$ to find the value of $x$

Q. Three circles each of radius $3.5$ cm are drawn in such a way that each of them touches the other two. Find the area enclosed between these circles.

1. $3.932c{m}^2$
2. $1.966c{m}^2$
3. $19.251c{m}^2$
4. $21.217c{m}^2$

Q. Construct an isosceles triangle whose base is $8cm$ and altitude $4cm$ and then another triangle whose sides are $1\frac{1}{2}$ times the corresponding sides of the isosceles triangle.

Q. In the given figure, ABCD is a quadrilateral with $\angle 1=\angle 2$ and $\angle 3=\angle ABC$ and $\angle 4=\angle BAD$. Also, AP and BQ are produced to points E and F, respectively. Then, we can say that

1. $\Delta OFE$ is an equilateral triangle
2. $\Delta OFE$ is a scalene triangle
3. $\Delta OFE$ is an isosceles triangle
4. Can't be determined

Q. A vertical lamp-post, 6 m high, stands at a distance of 2 m from a wall, 4 m high. A 1.5-m-tall man starts to walk away from the wall on the other side of the wall, in line with the lamp-post.The maximum distance to which the man can walk remaining in the shadow is

1. $\frac{5}{2}m$
2. $\frac{3}{2}m$
3. $4m$
4. none of these

Q. The area of the triangle whose vertices are $A(1,1),B(7,3)$ and $C(12,2)$ is

1. $8$ square units
2. $25$ square units
3. $12$ square units
4. $16$ square units

Q. For what value of $k(k>0)$ is the area of the triangle with vertices $(-2,5),(k,-4)$ and $(2k+1,10)$ equal to $53$ square units?

Q. The angles of a triangle are in the ratio $2:3:7$ and the radius of the circumscribed circle is $10$cm. The length of smallest side is?

1. $2$ cm
2. $5$ cm
3. $7$ cm
4. $10$ cm

Q. From the top of a hill, the angles of depression of two consecutive kilometer stones due east are found to be ${30}^o$ and ${45}^o$. Find the height of the hill

1. $1.365$km
2. $1.5$km
3. $1.7$km
4. $1.1$km

Q. In given figure, find $\angle F$

Q. $△$ABC is a right angled triangle, right angled at B. BD is a perpendicular to AC as shown in the figure. If AD:DC = 1:2, then $\frac{{AB}^2}{{BC}^2}$ is

Q. The shadow of a tower standing on a level plane is found to be 50 m longer when sun's elevation is ${60}^0$ than when it is ${30}^{\circ }$. The height of the tower is $25\sqrt{p}$, then $p$ is