Similar Triangles

Q.

If an equilateral triangle is drawn inside a circle such that the circle is the circum-circle of the triangle, find the relation between the length of the triangle and the radius of the circle.

1. r/3

2. 3r

3. r/√3

4. √3 x r

Q. In the given figure, $AD=3cm,AE=5cm,BD=4cm,CE=4cm,CF=2cm,BF=2.5cm$, then find the pair of parallel lines and hence their lengths.

Q.

P and Q are points on sides AB and AC respectively of $△ABC$. If AP = 3 cm, PB = 6cm. AQ = 5 cm and QC = 10 cm, show that BC = 3PQ.

Q.

In a given $\Delta ABC,DE\parallel BC$ and $\frac{AB}{DB}=\frac{3}{5}$. If AC = 5.6, find AE. [3 MARKS]

Q. $△ABC$ and $△DEF$ are similar such that $2AB=DE$ and $BC=8cm$. Find $EF$.

1. $16cm$
2. $12cm$
3. $8cm$
4. $4cm$

Q.

The perimeters of two similar triangles are 25 cm and 15 cm respectively. If one side of first triangle is 9 cm, the corresponding side of the other triangle is

1. 4.5 cm

2. 5.6 cm

3. 3 cm

4. 5.4 cm

Q.

Question 126 (ii)

If $\Delta$ is an operation, such that for integers a and b. We have $a\Delta b=a\times b-2\times a\times b+b\times b(-a)\times b+b\times b,$ then find
$(-7)\Delta (-1)$
Also, show that $(-7)\Delta (-1)\ne (-1)\Delta (-7)$

Q.

The ratio between the altitudes of two similar triangles is 3 : 5; write the ratio between their : (i) medians. (ii) perimeters. (iii) areas.

Q.

The diagonal BD of a parallelogram ABCD intersects the segment AE at the point F, where E is any point on the side BC. Prove that DF × EF = FB × FA.

Q.

Triangle ABC is similar to triangle PQR. If AD and PM are altitudes of the two triangles, then $\frac{AB}{PQ}=\frac{AD}{PM}$

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

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

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

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

Q.

D, E, F are the midpoints of the sides BC, CA and AB respectively of $△$ ABC. Then $△$ DEF is congruent to triangle –

1. ABC

2. AEF

3. BFD, CDE

4. AFE, BFD, CDE

Q. If $\Delta ABC\sim \Delta PQR$, $h_1$ and $h_2$ are the heights of $\Delta ABC$ and $\Delta PQR$ respectively AB : PQ = 3 : 5 and $h_2=12cm$, then the value of $h_1$ is

1. 7.2 cm
2. 4. 8 cm
3. 12 cm
4. 10 cm

Q.

In figure, DEFG is a square and $\angle BAC={90}^o$, Prove that
(i)$△AGF\sim △DBG$
(ii)$△AGF\sim △EFC$
(iii)$△DBG\sim △EFC$
(iv)${DE}^2=BD\times EC$

Q.

The function $f\left(x\right)=2x^3-3x^2-12x+4$ has

1. No maxima and minima

2. One maximum and one minimum

3. Two maxima

4. Two minima

Q.

If one angle of a triangle is equal to one angle of the other triangle and the sides including these angles are proportional, then the two triangles are similar.

1. True

2. False

Q.

Two chords AB and CD of a circle intersect at a point P outside the circle. Prove that

(1) $\Delta PAC\sim \Delta PDB$

(2) $PA.PB=PC.PD$.

Q.

All rectangles are __________ to each other. (Similar/ Not Similar)

Q. ∆ABD is a right triangle right-angled at A and AC ⊥ BD. Show that

(i) AB² = BC . BD
(ii) AC² = BC . DC
(iii) AD² = BD . CD
(iv) $\frac{{\mathrm{AB}}^2}{{\mathrm{AC}}^2}=\frac{\mathrm{BD}}{\mathrm{DC}}$

Q.

Two triangles are said to be similar, if the corresponding sides are ________ and the corresponding angles are ________.

1. proportional, equal

2. equal, proportional

3. equal, equal

4. proportional, proportional

Q. In the given figure, we have AB || CD || EF. If AB = 6 cm, CD = x cm, EF = 10 cm, BD = 4 cm and DE = y cm, Calculate the values of x and y.

Q.

If two or more than two triangles be equiangular and the ratios of their corresponding sides are equal, then the triangles are called___triangles.

Q.

The perimeter of two similar triangles $\mathrm{ABC}$ and $\mathrm{LMN}$ are $60$ $cm$and $48$ $cm$ respectively. If $\mathrm{LM}=8$ $cm$, then what is the length of $\mathrm{AB}$ ?

Q. As shown in the given figure, two poles of height 8 m and 4 m are perpendicular to the ground. If the length of shadow of smaller pole due to sunlight is 6 m then how long will be the shadow of the bigger pole at the same time ?

Q. The perimeters of two similar triangles are 30cm and 20cm respectively. If one side of the first triangle is 12cm determine the corresponding side of the second d triangle.

Q. In the given figure, MN∥BC and AM : MB = 1 : 2.
Find$\frac{\mathrm{area}\left(△\mathrm{AMN}\right)}{\mathrm{area}\left(△\mathrm{ABC}\right)}$

Q. Question 7
In the figure, DE || QR and AP and BP are bisectors of $\angle EABand\angle RBA$, respectively. Find $\angle APB$.

Q.

In the adjoining figure, $\angle ABC={90}^{\circ }$ and $BD\perp AC$. If AB = 5.7 cm, BD = 3.8 cm, CD = 5.4 cm, then find the length of BC.

1. 6.1 cm

2. 7.1 cm

3. 4.1 cm

4. 8.1 cm

Q.

If one of the diagonal of a trapezium divides the other in the ratio 2:1 prove that one of the parallel sides is twice the other

Q.

In the given figure, $A–D–C$ and $B–E–C$ , $DE\parallel AB$.

If $AD=5,DC=3,BC=6.4$ then Find $BE$.

Q.

(i) In Fig. 7.70, if AB ||CD, find the value of x.
(ii) In Fig. 7.71, if AB || CD, find the values of x.

(iii) In Fig. 7.72 AB||CD, If OA =3x-19, OB=x-4, OC=x-3 and OD= 4, find x.