In $\Delta ABC$,$AB = AC$ and $AD$ is an altitude to the base $BC$. Prove that $D$ is the midpoint of $BC$.
(N/A) Given: In $\Delta ABC$,$AB = AC$ and $AD \perp BC$.
To prove: $BD = DC$.
Proof: Consider $\Delta ABD$ and $\Delta ACD$.
$1$. $AB = AC$ (Given).
$2$. $\angle ADB = \angle ADC = 90^{\circ}$ (Since $AD$ is an altitude).
$3$. $AD = AD$ (Common side).
By the $RHS$ (Right angle-Hypotenuse-Side) congruence criterion,$\Delta ABD \cong \Delta ACD$.
Since the triangles are congruent,their corresponding parts are equal $(CPCT)$.
Therefore,$BD = DC$.
This implies that $D$ is the midpoint of $BC$.
To prove: $BD = DC$.
Proof: Consider $\Delta ABD$ and $\Delta ACD$.
$1$. $AB = AC$ (Given).
$2$. $\angle ADB = \angle ADC = 90^{\circ}$ (Since $AD$ is an altitude).
$3$. $AD = AD$ (Common side).
By the $RHS$ (Right angle-Hypotenuse-Side) congruence criterion,$\Delta ABD \cong \Delta ACD$.
Since the triangles are congruent,their corresponding parts are equal $(CPCT)$.
Therefore,$BD = DC$.
This implies that $D$ is the midpoint of $BC$.
Explore More
Similar Questions
$\angle ABD$ and $\angle ACE$ are exterior angles of $\Delta ABC$. If $\angle ABD = 140^{\circ}$ and $\angle ACE = 80^{\circ}$,then find $\angle A$. (in $^{\circ}$)
Medium
View SolutionIn $\Delta ABC$,$\angle A : \angle B : \angle C = 5 : 7 : 8$. Find the measure of each angle of $\Delta ABC$.
Medium
View Solution$\angle ABD$ and $\angle ACE$ are exterior angles of $\Delta ABC$. If $\angle ABD = 110^{\circ}$ and $\angle ACE = 150^{\circ}$,then $\angle A$ is: (in $^{\circ}$)
Medium
View SolutionIn a huge park,people are concentrated at three points (see Fig) :
$A$ : where there are different slides and swings for children,
$B$ : near which a man-made lake is situated,
$C$ : which is near to a large parking and exit. Where should an ice cream parlour be set up so that the maximum number of persons can approach it?
(Hint: The parlour should be equidistant from $A$,$B$,and $C$.)
$A$ : where there are different slides and swings for children,
$B$ : near which a man-made lake is situated,
$C$ : which is near to a large parking and exit. Where should an ice cream parlour be set up so that the maximum number of persons can approach it?
(Hint: The parlour should be equidistant from $A$,$B$,and $C$.)
Medium
View Solution$S$ is any point on side $QR$ of a $\triangle PQR$. Show that: $PQ + QR + RP > 2 \, PS$.
Medium
View SolutionVedclass Products
For Students
Vedclass Test Series
Mock tests in real JEE/NEET style with performance analysis. 5-day free trial.
Start Free TrialFor Teachers
Exam Paper Generator
Generate Set A/B/C/D exam papers from 7.5L+ questions in 2 minutes. 3 chapters free.
Try FreeFor Institutes
Online Exam Module
Live online exams with unlimited students, 360° analytics & white-label branding.
See Demo