Relation between Areas and Sides of Similar Triangles

Q. In a right angle triangle $\Delta ABC$, a perpendicular BD is drawn on to the largest side from the vertex B. 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 triangle $ABC,\angle CBA={72}^{\circ },E$ is the midpoint of side $AC$, and $D$ is a point on side $BC$ such that $2BD=DC$; $AD$ and $BE$ intersect at $F$. The ratio of the area of $△BDF$ to the area of quadrilateral $FDCE$ is

1. $\frac{2}{5}$
   
2. $\frac{1}{4}$
   
3. $\frac{1}{5}$
   
4. $\frac{1}{3}$
   

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

Q. In a right triangle $\Delta ABC$ with right angle at B, a perpendicular BD is drawn from B to the hypotenuse. If AC = 2AB, find the area of $\Delta ADB$.
Given: Area of $\Delta ABC$ = 5 sq units

1. 0.5 sq. unit
2. 1 sq. unit
3. 1.25 sq. unit
4. 5 sq. unit

Q. Find the ratio of the areas of two similar triangles $\Delta ABC$ and $\Delta PQR$ shown in the figure.

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

Q.

The line segment XY is parallel to side AC of Δ ABC and it divides the triangle into two parts of equal areas. Find the ratio $\frac{BX}{AB}$.

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

Q.

If $△ABC\sim △$ DEF such that AB = 12 cm and DE = 14 cm. Find the ratio of areas of $△$ ABC and $△$ DEF.

1. $\frac{25}{49}$

2. $\frac{49}{16}$

3. $\frac{36}{49}$

4. $\frac{49}{9}$

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. $\sqrt{2}:1$

2. $2:1$

3. $4:1$

4. $8:1$

Q.

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

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

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

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

4. All of the above.

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