Ratio of Areas of Two Similar Triangles

Q.

Sides of a triangle are in the ratio of $12:17:25$ and its perimeter is$540cm$. Find its area.

Q.

$If△ABC\sim △DEFsuchthatAB=6cmandDE=7cm.Findtheratioofareasof△ABCand△DEF.$

1. 49/36

2. 3/12

3. 36/49

4. 6/7

Q. Question 1
Let $\Delta ABC\sim \Delta DEF$ and their areas be, respectively, $64c{m}^2$ and $121c{m}^2$. If EF = 15.4 cm, find BC.

Q.

In given figure $△$ ABC and $△$ DEF are similar, BC=3cm, EF=4cm, and area of triangle ABC=54$c{m}^2$. Find the area of $△$ DEF. (Note: The given figures are not to scale.)

__ $c{m}^2$

Q. In the given figure $\angle ABC={90}^0$ and BD ⊥ AC. If $BD=8cm$ and $AD=4cm$, find $CD$.

Q.

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

1. ${\left(\frac{AB}{BC}\right)}^2$

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

3. ${\left(\frac{BD}{DC}\right)}^2$

4. ${\left(\frac{AD}{BD}\right)}^2$

Q.

$△ABC$ is an isosceles triangle in which AB = AC = 13 cm. If area of $△ADC$ is $169m^2$, then area of $△EFB$ is equal to

1. $196c{m}^2$

2. $324c{m}^2$

3. $169c{m}^2$

4. $396c{m}^2$

Q.

If $△$ABC and $△$PQR are two similar triangles shown in the figure. AM and PN are the medians on $△$ABC and $△$PQR respectively. The ratio of areas of $△$ABC and $△$PQR is 9:25. If AM = PO = 5 cm. Find the value of 3ON.

Q.

If triangle ABC is an isosceles triangle in which AB = AC = 13 cm, then find the value of $\frac{\text{area of}\Delta ADC}{\text{area of}\Delta EFB}.$ (upto two decimal places)

1. 0.52

Q. In a right angled 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 ratios 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. ABC is an equilateral triangle of side 2a. Find each of its altitudes.

1. $\sqrt{2}a$
2. $\sqrt{3a}$
3. $\sqrt{3}a$
4. $\sqrt{2a}$

Q.

In a right angled $△$ABC, a perpendicular BD is drawn on to the largest side from the opposite vertex. Which of the following does NOT give the ratios of the areas of $△$ABD and $△$ BCD?

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

2. ${\left(\frac{AB}{BC}\right)}^2$

3. ${\left(\frac{BD}{DC}\right)}^2$

4. ${\left(\frac{AD}{BD}\right)}^2$

Q. Find the ratio of the areas of two triangles shown in the figure. If $\Delta ABC$ ~$\Delta PQR$ ​​​​​.

1. ${\left(\frac{AB}{PQ}\right)}^2$
2. ${\left(\frac{BC}{QR}\right)}^2$
3. ${\left(\frac{AC}{PR}\right)}^2$
4. $Allofthese$

Q. $\Delta ABC\sim \Delta PQR$ and $ar\left(\Delta ABC\right)=4ar\left(\Delta PQR\right)$. If $BC=12cm$, find $QR$.

Q. Find the ratio of the areas of two similar triangles shown in the figure.

1. ${\left(\frac{AB}{PQ}\right)}^2$
2. ${\left(\frac{BC}{QR}\right)}^2$
3. ${\left(\frac{AC}{PR}\right)}^2$
4. $Allofthese$

Q.

D and E are points on the sides AB and AC respectively of a $△$ABC such that DE || BC. Which of the following statement is true?

(i) $△$ ADE ~ $△$ ABC

(ii) $\frac{areaof△ADE}{areaof△ABC}$ = $\frac{A{D}^2}{A{B}^2}$

(iii) $\frac{areaof△ADE}{areaof△ABC}$ = $\frac{A{B}^2}{A{C}^2}$

1. only (i)

2. only (i) and (ii)

3. only (iii)

4. all (i) , (ii) and (iii)

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 ratios 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 given figure $△$ ABC and $△$ DEF are similar, BC = 3cm, EF = 4cm, and area of triangle ABC = 54$c{m}^2$. The area of $△$ DEF is ___ ${cm}^2$.

Q.

If D, E, F are the midpoints of sides BC, CA and AB of $△$ ABC, then the ratio of the area of $△$ DEF to $△$ ABC is:

1. 1: 4

2. 1: 2

3. 2: 3

4. 4: 5

Q.

In the figure, $△ABC$ and $△DBC$ are on the same base $BC$. $AD$ and $BC$ intersect at $O$. Prove that $\frac{area(∆ABC)}{area(∆DBC)}=\frac{AO}{DO}$.

Q.

$Inthefigure,givenalongside,PBandQAareperpendicularstothelinesegmentAB.IfPO=3cm,QO=4.5cmandareaof△POB=60c{m}^2,findtheareaof△QOA.$

1. 135 cm²

2. 130 cm²

3. 270 cm²

4. 70 cm²

Q.

In the figure, PB and QA are perpendicular to segment AB. If OA = 5 cm, PO = 7cm and area ($\Delta$QOA) = 150 $c{m}^2$, find the area of $\Delta$POB.

1. $233c{m}^2$

2. $294c{m}^2$

3. $300c{m}^2$

4. $420c{m}^2$

Q. The perimeters of two similar triangles $ABC$ and $PQR$ are $32cm$ and $24cm$ respectively. If $PQ=12cm$, find $AB$.

Q.

In the given figure, $△ABC\sim △$PQR and AM and PN are medians of $△$ABC and $△$PQR respectively. Then $\frac{Areaof△ABC}{Areaof△PQR}$ = _____

1. $\frac{{AB}^2}{{PQ}^2}$

2. $\frac{{AM}^2}{{PN}^2}$

3. (a) and (b) above.
4. $\frac{{PQ}^2}{{QR}^2}$

Q. In Fig., $D$ is a point on side $BC$ of $△ABC$ such that $\frac{BD}{CD}=\frac{AB}{AC}.$ Prove that $AD$ is the bisector of $\angle BAC$.

Q. The areas of triangles ABD and ADC are 4 $c{m}^2$ and 12 $c{m}^2$ respectively. Find the ratio of their bases.

Q.

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

1. ${\left(\frac{BD}{DC}\right)}^2$

2. ${\left(\frac{AB}{BC}\right)}^2$

3. ${\left(\frac{AB}{AC}\right)}^2$

4. ${\left(\frac{AD}{BD}\right)}^2$

Q.

$ABCD$ is a square. Equilateral triangles $ACF$ and $ABE$ are drawn on the the diagonal $AC$ and side $AB$ respectively. Find area of $△ACF$ : area of $△ABE$.

1. $2:1$

2. $4:1$

3. $\sqrt{2}:1$

4. $8:1$

Q.

$\text{In the given figure,}△ABC\sim △PQR.\text{Then,}\frac{areaof△ABC}{areaof△PQR}\text{equals}$

1. $\frac{{AB}^2}{{PQ}^2}$

2. $\frac{{BC}^2}{{QR}^2}$

3. $\frac{{AC}^2}{{PR}^2}$

4. All of the above.

Q.

Find the ratio of area of the equilateral triangles whose sides are 3 and 4 units.

1. 9 : 25

2. 9: 16

3. 16: 25

4. None of the above