Question
In the given figure, $\angle\text{ABC}=90^\circ$ and $\text{BD} \perp\text{AC.}$ If AB = 5.7cm, BD = 3.8cm and CD = 5.4cm, find BC.
✓
Answer
We have, $\angle\text{ABC}=90^\circ$ and $\text{BD}\perp\text{AC}$
In $\triangle\text{ABC}$ and $\triangle\text{BDC}$
$\angle\text{ABC}=\angle\text{BDC}$ [Each 90°]
$\angle\text{C}=\angle\text{C}$ [Commom]
Then, $\triangle\text{ABC}\sim\triangle\text{BDC}$ [By AA similarity]
$\therefore\frac{\text{AB}}{\text{BD}}=\frac{\text{BC}}{\text{DC}}$ [Corresponding parts of similar $\triangle$ are proportional]
$\Rightarrow\frac{5.7}{3.8}=\frac{\text{BC}}{5.4}$
$\Rightarrow\text{BC}=\frac{5.7}{3.8}\times8.1\text{cm}$
In $\triangle\text{ABC}$ and $\triangle\text{BDC}$
$\angle\text{ABC}=\angle\text{BDC}$ [Each 90°]
$\angle\text{C}=\angle\text{C}$ [Commom]
Then, $\triangle\text{ABC}\sim\triangle\text{BDC}$ [By AA similarity]
$\therefore\frac{\text{AB}}{\text{BD}}=\frac{\text{BC}}{\text{DC}}$ [Corresponding parts of similar $\triangle$ are proportional]
$\Rightarrow\frac{5.7}{3.8}=\frac{\text{BC}}{5.4}$
$\Rightarrow\text{BC}=\frac{5.7}{3.8}\times8.1\text{cm}$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
If in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio (or proportion) and hence the two triangles are similar:
In the given figure, from the top of a solid cone of height $12 \ cm$ and base radius $6 \ cm ,$ a cone of height $4 \ cm$ is removed by a plane parallel to the base. Find the total surface area of the remaining solid. Use $\left(\pi=\frac{22}{7}\right.$ and $\left.\sqrt{5}=2.236\right)$.
→A hemispherical bowl of internal radius 9cm is full of liquid. The liquid is to be filled into cylindrical shaped bottles each of radius 1.5cm and height 4cm. How many bottles are needed to empty the bowl?
If $\sin3\text{A}=\cos(\text{A}-26^\circ),$ where 3A is an acute angle, then find the value of A.
→If $\cot\theta=\frac{1}{\sqrt{3}},$ find the value of $\frac{1+\cos^2\theta}{2-\sin^2\theta}$.
→State the two properties which are necessary for given two triangles to be similar.
In a school, students thought of planting trees in and around the school to reduce air pollution. It was decided that the number of trees, that each section of each class will plant, will be the same as the class, in which they are studying, e.g., a section of Class I will plant $1$ tree, a section of Class II will plant $2$ trees and so on till Class XII. There are three sections of each class. How many trees will be planted by the students?
→The difference of squares of two numbers is $180.$ The square of the smaller number is $8$ times the larger number. Find two numbers.
→Find the roots of the following equations, if they exist, by applying the quadratic formula:$4\sqrt3\text{x}^2+5\text{x}-2\sqrt3=0$
→Using Euclid's algortihm, find the HCF of:
504 and 1188
504 and 1188