AAA Similarity

Q.

The length of an altitude of an equilateral triangle having side length of 6 cm is $2\sqrt{6}$ cm.

1. True

2. False

Q.

Consider the following parallelogram. Find the values of the unknown x, y, z

Q. Question 9
In given figure, ABC is a triangle right angled at B and $BD\perp AC$. If AD =4 cm and CD = 5cm, then find BD and AB.

Q. In a ∆ABC, D and E are points on AB and AC respectively such that DE || BC. If AD = 2.4 cm, AE = 3.2 cm, DE = 2 cm and BC = 5 cm, find BD and CE.

Q.

In Fig. 7.144, $\Delta ABC$ is right angled at C and $DE\perp AB$. Prove that $\Delta ABC\sim \Delta ADE$ and hence find the lengths of AE and DE.

Q. In a trapezium ABCD, AB|| DC and DC = 2AB, FE drawn parallel to AB cuts AD in F and BC in E such that 4BE = 3EC. Diagonal DB intersects EF at G. Prove that 7FE = 10 AB. [4 MARKS]

Q.

Through the mid-point M of the side CD of a parallelogram ABCD, the line BM is drawn intersecting diagonal AC in L and AD produced in E. Prove that : EL = 2 BL.

Q.

If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then the other two sides are divided in the same ratio.

Q. In the given figure, RQ and TP are perpendicular to PQ, also $TS\perp PR$ prove that ST. RQ = PS. PQ.

Q.

In the given figure, $DB\perp BC,DE\perp ABandAC\perp BC.$

Prove that $\frac{BE}{DE}=\frac{AC}{BC}$.

Q. In a ∆ABC, D and E are points on the sides AB and AC respectively such that DE || BC.

(i) If AD = 6 cm, DB = 9 cm and AE = 8 cm, find AC.
(ii) If $\frac{\mathrm{AD}}{\mathrm{DB}}=\frac{3}{4}$ and AC = 15 cm, find AE.
(iii) If $\frac{\mathrm{AD}}{\mathrm{DB}}=\frac{2}{3}$ and AC = 18 cm, find AE.
(iv) If AD = 4, AE = 8, DB = x − 4, and EC = 3x − 19, find x.
(v) If AD = 8 cm, AB = 12 cm and AE = 12 cm, find CE.
(vi) if AD = 4 cm, DB = 4.5 cm and AE = 8 cm, find AC.
(vii) If AD = 2 cm, AB = 6 cm, and AC = 9 cm, find AE.
(viii) If $\frac{\mathrm{AD}}{\mathrm{BD}}=\frac{4}{5}$ and EC = 2.5 cm, find AE.
(ix) If AD = x, DB = x − 2, AE = x + 2 and EC = x − 1, find the value of x.
(x) If AD = 8x − 7, DB = 5x − 3, AE = 4x − 3 and EC = (3x − 1), find the value of x.
(xi) If AD = 4x − 3, AE = 8x − 7, BD = 3x − 1 and CE = 5x − 3. find the volume x.
(xii) If AD = 2.5 cm, BD = 3.0 cm and AE = 3.75 cm find the length of AC.

Q. If ABC and DEF are similar triangles such that ∠A = 47° and ∠E = 83°, then ∠C =

(a) 50°
(b) 60°
(c) 70°
(d) 80°

Q. Theorem: If a line parallel to a side of a triangle intersects the remaining sides in two distinct points, then the line divides the sides in the same proportion.

Q.

In a triangle ABC and DE parallel to BC , if AD= 1.5, AE=1cm and DB =3cm, find EC

Q.

In the figure, PQRS is a parallelogram with PQ = 16 cm and QR = 10 cm. L is a point Q on PR such that RL: LP = 2:3. QL produced meets RS at M and PS produced at N. Find the lengths of PN and RM.

Q. ABCD is a parallelogram in which P is the midpoint of DC and Q is point on AC such that CQ=1/4 AC . If produced PQ meets at BC at R then prove that R is the midpoint of BC.

Q.

The shadow of a 5-m-long stick is 2 m long. At the same time, the length of the shadow of a 12.5-m-high tree is

(a) 3 m

(b) 3.5 m

(c) 4.5 m

(d) 5 m

Q. Question 181
In parallelogram ABCD, the angle bisector of $\angle A$ bisects BC. Will angle bisector of B also bisect AD? Give reason.

Q.

In $△ABC$, DE is parallel to base BC, with D on AB and E on AC. If $\frac{AD}{DB}=\frac{2}{3}$, find $\frac{BC}{DE}$

Q.
In the given figure, ABC is a triangle, DE is parallel to BC and $\frac{AD}{DB}=\frac{3}{2}$
(i) Determine the ratios $\frac{AD}{AB}and\frac{DE}{BC}.$
(ii) Prove that $\Delta DEF$ is similar to $\Delta CBF$ Hence, find $\frac{EF}{FB}$.
(iii) What is the ratio of the areas of $\Delta DFEand\Delta BFC$? [4 MARKS]

Q.

CD and GH are respectively the bisectors of ∠ACB and ∠EGF such that D and H lie on sides AB and FE of ΔABC and ΔEFG respectively. If ΔABC ∼ ΔFEG, Show that:

(i)

(ii) ΔDCB ∼ ΔHGE

(iii) ΔDCA ∼ ΔHGF

Q. Question 8
ABC and DBC are two triangles on the same base BC such that A and D lie on the opposite sides of BC, AB = AC and DB = DC. Show that AD is the perpendicular bisector of BC.

Q.

In the figure, given below, ABCD is a parallelogram. P is a point on BC such that BP : PC = 1 : 2. DP produced meets AB produced at Q. Given the area of triangle CPQ = 20 $c{m}^2$.
Calculate : (i) area of triangle CDP, (ii) area of parallelogram ABCD.

Q. Question 177
A diagonal of a parallelogram bisects an angle. Will it also bisect the other angle? Give reason.

Q. Question 32
If PQRS is a parallelogram, then $\angle P-\angle R$ is equal to
a) ${60}^{\circ }$
b) ${90}^{\circ }$
c) ${80}^{\circ }$
d) $0^{\circ }$

Q. Question 8
E is a point on the side AD produced of a parallelogram ABCD and BE intersect CD at F. Show that $\Delta ABE\sim \Delta CFB$.

Q.

Question 8
A street light bulb is fixed on a pole 6 m above the level of the street. If a woman of height 1.5 m casts a shadow of 3m, then find how far she is away from the base of the pole.

Q. Question 1
In given figure, if $\angle A=\angle C$, AB = 6cm, BP = 15cm, AP = 12cm and CP = 4cm, then find the lengths of PD and CD.

Q.

If two angles of two triangles are congruent to each other, then the third angles will be congruent to each other.

Using the above statement, choose which two congruency criteria eventually becomes same.

1. AAS and SAS

2. SAS and ASA

3. AAS and ASA

4. No two congruency criteria are same

Q.

A tower of height 24 m casts a shadow 50 m and at the same time, a girl of height 1.8 m casts a shadow. Find the length of her shadow.

1. 3 m

2. 3.25 m

3. 3.5 m

4. 3.75 m