Question

Prove that in an isosceles triangle, the median to the base is also perpendicular.

Answer 1

Hint: Properties which must be known while solving this question are as follows: -
i) SSS (side side side ) congruence rule of triangle- In this two triangles are said to satisfy SSS congruent rule in which all three sides of one triangle are equal to the corresponding sides of another triangle.
ii) Median divides the line into two equal parts.
iii) Isosceles triangle has two equal base angles.

Complete step-by-step answer:
To prove: In an isosceles triangle median to the base is also perpendicular.
Proof: Let ABC be an isosceles triangle having AD as a median which divides side BC into two equal parts (BD=DC) making D as a midpoint, below is the figure mentioning that.

To proof median perpendicular we have to proof $$\mathrm{\angle }ADB=\mathrm{\angle }ADC$$
In $\mathrm{\Delta }ADB$ and $\mathrm{\Delta }ADC$
$AB=AC$ (Isosceles triangle given)
$BD=DC$ (Median divides BC into two equal parts)
$AD=AD$ (Common)
 (SSS congruent rule)
$\therefore$ $\mathrm{\angle }ADB=\mathrm{\angle }ADC$ (By corresponding parts of congruent triangles)
Now let $\mathrm{\angle }ADB=ADC=x^{\circ }$
$\therefore$ $\mathrm{\angle }ADB+\mathrm{\angle }ADC={180}^{\circ }$ (Sum of angles on a straight line is ${180}^{\circ }$ )

$x^{\circ }+x^{\circ }={180}^{\circ }$
Solving for $x$,
$2x^{\circ }={180}^{\circ }$
On simplifying the above equation, we get
$x^{\circ }={90}^{\circ }$

$\therefore$ $\mathrm{\angle }ADB=\mathrm{\angle }ADC={90}^{\circ }$

Hence it is proved that median to the base is also perpendicular.

Note:
Alternate method: To prove triangles congruency one can also use SAS (side angle side ) rule.
For example: $\mathrm{\Delta }ADB$ and $\mathrm{\Delta }ADC$
$AB=AC$ (Isosceles triangle given)
$\mathrm{\angle }ABC=\mathrm{\angle }ACD$ (Isosceles base angles equal)
$BD=DC$ (Median divides BC into two equal parts)
 (SAS rule of congruency)
There are more congruent properties of a triangle:-
AAS (angle angle side) congruency – It states that when two angles and non included side of one triangle is equal to the corresponding two angles and non included side of another triangle then it is said to be AAS property of congruency.

ASA (angle side angle) congruency – It states that if two angles including side of one triangle are equal to corresponding two angles and including side of another triangle then it is said to be ASA congruency.