Question
ABC is a triangle right angled at B and $\text{BD}\perp\text{AC}.$ If AD = 4cm, and CD = 5cm, find BD and AB.
✓
Answer
In $\triangle\text{ABC},$$\angle\text{ABC}=90^\circ$ [Given]
$\text{BD}\perp\text{AC}$ [Hypotenuse]
$\therefore\text{BD}^2=\text{DA}\times\text{DC}$
$\Rightarrow\text{BD}^2=4\times5$
$\Rightarrow\text{BD}=2\sqrt{5}\text{cm}$
In right angled $\triangle\text{BDA},$
$\text{BD}\perp\text{AC}$ [Given]
$\therefore\angle\text{BDA}=90^\circ$
$\Rightarrow\text{AB}^2=\text{AD}^2+\text{BD}^2$ [by Pythagoras theorem]
$\Rightarrow\text{AB}^2=4^2+(2\sqrt{5})^2$
$\Rightarrow\text{AB}^2=16+20$
$\Rightarrow\text{AB}^2=36$
$\Rightarrow\text{AB}=6\text{cm}$
Hence proved.
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
Show graphically that each of the following given systems of equations is inconsistent, i.e., has no solutions:
2x + y = 2, 2x + y = 6
2x + y = 2, 2x + y = 6
A train, travelling at a uniform speed for 360km, would have taken 48 minutes less to travel the same distance if its speed were 5km/hr more. Find the original speed of the train.
A regular hexagon is inscribed in a circle. If the area of hexagon is $24\sqrt{3}\text{cm}^2,$ find the area of the circle. $\big(\text{Use }\pi=3.14\big)$
→Solve for x and y:
x + y = 5xy,
3x + 2y = 13xy $(\text{x}\neq0,\ \text{y}\neq0).$
→x + y = 5xy,
3x + 2y = 13xy $(\text{x}\neq0,\ \text{y}\neq0).$
Solve the following quadratic equations by factorization:
$7\text{x}+\frac{3}{\text{x}}=35\frac{3}{5}$
→$7\text{x}+\frac{3}{\text{x}}=35\frac{3}{5}$
Name the quadrilateral formed, if any, by the following points, and given reason for your answers:
A(-3, 5), B(3, 1), C(0, 3), D(-1, -4).
A(-3, 5), B(3, 1), C(0, 3), D(-1, -4).
A chord 10cm long is drawn in a circle whose radius is $5\sqrt{2}\text{cm}.$ Find area of both the segments.
→In a $\triangle\text{ABC},\angle\text{B}=90^\circ,\angle\text{AB}=12\text{cm}$ and BC = 5cm.
Find:
→Find:
- $\cos\text{A}$
- $\text{cosec}\text{A}$
- $\cos\text{C}$
- $\text{cosec}\text{C}.$
A train travels 180km at a uniform speed. If the speed had been 9km/hr more, it would have taken 1 hour less for the same journey. Find the speed of the train.
Show graphically that each of the following given systems of equations has infinitely many solutions:
x - 2y = 5, 3x - 6y = 15
x - 2y = 5, 3x - 6y = 15