Criteria for Similarity of Triangles

Q.

In the figure AB = BC and AD is perpendicular to CD. Prove that : AC² = 2BC. DC.

Q.

In the figure below, if the line segment ST is parallel to line segment QR such that $\frac{PS}{SQ}=\frac{PT}{TR}$. The provided data is not sufficient to prove that triangles PQR and PST are similar.

1. True

2. False

Q. AD, BE and CF, the altitudes of $\Delta ABC$ are equal. Prove that $\Delta ABC$ is an equilateral triangle

Q. BE and CF, the altitudes of $△$ABC are equal. Prove that $△$ABC is an equilateral triangle.

Q. In a triangle ABC, DE is parallel to BC if $\frac{AD}{DB}=$ $\frac{2}{3}$ and $AC=18$ cm find AE.

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 find the lengths of PA & PC.

Q. A man 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 the shadow of man after 5s.

Q. A plot of land is in the shape of a right angled isosceles triangle. The length of the hypotenuse is $50\sqrt{2}m$. The cost of fencing it at Rs. $3$ per mete will be

1. less than Rs. $300$
2. less than Rs. $400$
3. more than Rs. $500$
4. more than Rs. $600$

Q. If $\Delta DEF\cong \Delta BCA$ , then the angle of $\Delta BCA$ that corresponds to $\angle E$ is _______ and side $FD$ corresponds to side ________.

Q. In $\Delta ABC$, if $AD$ divides $BC$ in the ratio $m:n$, then area of $\Delta ABD:$ area of $\Delta ADC$ is _________.

1. $n:m$
2. $m:n$
3. $(m+1):n$
4. $m:(n+1)$

Q. The sides of an equilateral triangle ABC are 12 cm each. D is the foot of the perpendicular from A to BC and E is the mid-point of AD. BE is __________________.

1. 12 cm
2. $6\sqrt{2}cm$
3. $\sqrt{63}cm$
4. None of the above

Q. A man 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 the shadow of man after 5s.

Q.

In fig., $\angle$E $>$ $\angle$A and $\angle$C $>$ $\angle$D. Prove that AD $>$ EC.

Q. In the given figure, $AB$ and $CD$ are two parallel chords of a circle. if $BDE$ and $ACE$ are straight lines, intersecting at $E$, prove that $\Delta AEB$ is isosceles.

Q. In the given figure, PR > PQ and PS bisects $\angle$QPR. Prove that $\angle$PSR > $\angle$PSQ
.

Q. If $lo{g}_{e}4=1.3868$, then $lo{g}_{e}4.01=$

1. 1.3968
2. none of these
3. 1.3893
4. 1.3898

Q. ABCD is a parallelogram as shown in figure. If AB=2AD and P is mid-point of AB, then $\angle CPD$ is equal to

1. ${90}^{\circ }$
2. ${60}^{\circ }$
3. ${45}^{\circ }$
4. ${135}^{\circ }$

Q.

Q. In the given figure, PR > PQ and PS bisects $\angle$QPR. Prove that $\angle$PSR > $\angle$PSQ
.

Q. $ABCD$ is a trapezium with $AB\parallel DC$. A line parallel to $BD$ intersects $CD$ at $X$ and $BC$ at $Y$ such that $BD\parallel XY$. Prove that $ar\left(ADX\right)=ar\left(BDY\right)$.

Q. If $P$ is a point inside the scalene triangle $ABC$ such that $\Delta APB$, $\Delta BPC$ and $\Delta CPA$ have the same area, then $P$ must be:

1. In centre of $\Delta ABC$
2. Circum centre of $\Delta ABC$
3. Ortho centre of $\Delta ABC$
4. Centroid of $\Delta ABC$