In figure, Triangle ABC is a right angled triangle at B. Given th… - Vidyadip

Question

In figure, Triangle $\text{ABC}$ is a right angled triangle at $B$. Given that $AB = 9\ cm, AC = 15\ cm$ and $D, E$ are the mid$-$points of the sides $AB$ and $AC$ respectively, calculate
$i.$ The length of $BC$
$ii.$ The area of $\triangle\text{ADE}.$

✓

Answer

$i.$ In $\triangle\text{ABC},\angle\text{B}=90^\circ,$
By using Pythagoras theorem
$AC^2 = AB^2 + BC^2$
$\Rightarrow15^2=9^2+\text{BC}^2$
$\Rightarrow\text{BC}=\sqrt{15^2-9^2}$
$\Rightarrow\text{BC}=\sqrt{225-81}$
$\Rightarrow\text{BC}=\sqrt{144}=12\text{ cm}$
$ii.$ In $\triangle\text{ABC},$
$D$ and $E$ are mid$-$points of $AB$ and $AC$
$\therefore\text{DE}\|\text{BC},=\frac{1}{2}\text{BC} [$By mid$−$point theorem$]$
$\text{AD}=\text{DB}=\frac{\text{AB}}{2}=\frac{9}{2}=4.5\text{ cm}$
$[\therefore$ $D$ is the mid$−$point of $AB]$
Area of $\triangle\text{ADE}=\frac{1}{2}\times\text{AD}\times\text{DE}$
$=\frac{1}{2}\times4.5\times6$
$=13.5\text{ cm}^2$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "4 Marks Questions" from Quadrilaterals→Quadrilaterals — all question types→Maths STD 9 question papers→Gujarat Board STD 9 question papers→Gujarat Board question paper generator→

Similar questions

An equilateral triangle of side 9cm is inscribed in a circle. Find the radius of the circle.

A cylindrical bucket, $28\ cm$ in diameter and $72\ cm$ and high, is full of water. The water is emptied into a rectangular tank, $66\ cm$ long and $28\ cm$ wide. Find the height of the water level in the tank.

In the adjoining figure, D, E, F are the midpoints of the sides BC, CA and AB respectively, of $\triangle\text{ABC}.$ Show that $\angle\text{EDE}=\angle\text{A},\angle\text{DEF}=\angle\text{B}$ and $\angle\text{DFE}=\angle\text{C}.$

→

A square is inscribed in an isosceles right triangle so that the square and the triangle have one angle common. Show that the vertex of the square opposite the vertex of the common angle bisects the hypotenuse.

It being given that $\sqrt{3}=1.732,\sqrt{5}=2.236,\sqrt{6}=2.449$ and $\sqrt{10}=3.162,$ find to three places of decimal, the value of the following:
$\frac{3+\sqrt{5}}{3-\sqrt{5}}$

In the given figure, $ABCD$ is a cyclic quadrilateral. Find the value of $x$.

In the given figure, side $BC$ of $\triangle\text{ABC}$ is produced to point $D$ such that bisectors of $\angle\text{ACD}$ meet at a point $E$. If $\angle\text{BAC}=68^\circ,$ find $\angle\text{BEC}.$

In the given figure, O is the centre of the circle. If $\angle\text{ABD}=35^\circ$ and $\angle\text{BAC}=70^\circ,$find $\angle\text{ACB}.$

→

Draw the frequency polygon representing the above data without drawing the histogram.

In the adjoining figure, explain how one can find the breadth of the river without crossing it.