Question
$\text{AD}$ and $\text{BC}$ are equal perpendiculars to a line segment $\text{AB}$. If $\text{AD}$ and $\text{BC}$ are on different sides of $\text{AB}$ prove that $\text{CD}$ bisects $\text{AB}.$
✓
Answer
In $\triangle AOD$ and $\triangle BOC$,
$\angle \mathrm{AOD}=\angle \mathrm{BOC} \quad \ldots. ($vertically opposite angles$)$
$\angle \mathrm{DAO}=\angle \mathrm{CBO} \quad \ldots . .\left(\right.$ each $\left.90^{\circ}\right)$
$\text{AD}=\text{BC} \quad \quad \ldots. ($given$)$
$\therefore \triangle \mathrm{AOD} \cong \triangle \mathrm{BOC} \quad...($by $\text{AAS}$ congruence criterion$)$
$\Rightarrow \text{AO}=\text{BO} ...($c.p.c.t$.)$
$\Rightarrow O$ is the mid$-$point of $\text{AB}$. Hence, $\text{CD}$ bisects $\text{AB}$.
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
Construct a parallelogram ABCD in which $BC =5 cm, CD =3 cm$ and diagonal $BD =6 cm$.
→Through the vertex $S$ of a parallelogram $\text{PQRS},$ a line is drawn to intersect the sides $Qp$ and $QR$ produced at $M$ and $N$ respectively. Prove that $\frac{ SP }{ PM }=\frac{ MQ }{ QN }=\frac{ MR }{ SR }$
→Factorise the following by grouping the terms:$xy(a^2 + 1) + a(x^2 + y^2)$
→The breadth of a rectangular room is $2\ m$ less than its length. If the perimeter of the room is $14\ m,$ find it's dimensions.
→Evaluate the following without multiplying:$(999)^2$
→Solve the equations:
$\frac{x}{7}+\frac{y}{3}=5, \frac{x}{2}-\frac{y}{9}=6$
→$\frac{x}{7}+\frac{y}{3}=5, \frac{x}{2}-\frac{y}{9}=6$
Find the value of $x$ in the following $: \tan x = \sin 45^\circ \cos 45^\circ + \sin 30^\circ $
→If $x + y = 9, xy = 20$find: $x^2 - y^2.$
→How many liters of water will flow out of a pipe having a cross sectional area $6 \ cm^2$ in one hour, if the speed of water in the pipe is $30 \ cm/sec$?
→If $a + b = 11$ and $a^2+ b^2= 65;$ find $a^3+ b^3.$
→