Name: Congruent Circles and Arcs
Uploaded: 2019-07-19
Duration: 2 min 20 s
Description: A pair of bangles is one of the examples of congruent circles. Why are they important to us? What are the properties of congruent arcs? Watch this video to k...

Question

Prove that the median of an equilateral triangle is also its altitude.

Accepted Answer

We have to prove that the median of an equilateral triangle is also its altitude.

Let an equilateral triangle $PQR$ in which $PQ = QR = PR$ , and $PS$ is a median as shown below-

Since $PS$ is the median so, $QS = SR$ , and we know that the median bisects the vertex angle in an isosceles and equilateral triangle where the two adjacent sides are the same. Therefore,

$∠ QPS = ∠ RPS$

An altitude of a triangle is a line segment through a vertex and perpendicular to a line containing the base. So, we have to prove that $∠ PSQ = ∠ PSR = 90 °$ .

$∠ QPS = ∠ RPS = \frac{1}{2} ∠ QPR = \frac{1}{2} \times 60 °$ (Each angle of an equilateral triangle is $60 °$ )

$= 30 °$

By using angle sum property in $∆ PQS$ -

$∠ PQS + ∠ PSQ + ∠ QPS = 180 ° \Rightarrow 60 ° + ∠ PSQ + 30 ° = 180 ° \Rightarrow ∠ PSQ = 180 ° - 90 ° = 90 °$

So, $∠ PSQ = ∠ PSR = 90 °$ .

Hence, proved.

Prakash Tiwari and Ishita Joshi Solutions for Chapter: Congruence of Triangles, Exercise 5: Brain Teasers

Prakash Tiwari Mathematics Solutions for Exercise - Prakash Tiwari and Ishita Joshi Solutions for Chapter: Congruence of Triangles, Exercise 5: Brain Teasers

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