Name: Proof of Basic Proportionality theorem
Uploaded: 2019-07-19
Duration: 3 min 10 s
Description: When a line is drawn parallel to one side, it intersects the other two sides of a triangle in the same ratio. Watch this video to learn more about them.

Question

In an acute-angled triangle, express a median in terms of its sides.

Accepted Answer

Let $Δ A B C$ be acute angled triangle where $AD$ is its median with respect side $BC$ .
Construct $AE ⊥ BC$ .

Now since AD is the median,
$∴ BD = CD = 1 / 2 BC . . . . . . (1)$
In $ΔAED$ , applying Pythagoras Theorem,
$\Rightarrow AD ² = AE ² + DE ² \Rightarrow AE ² = AD ² - DE ² . . . . . . . (2)$

In $ΔAEB$ , applying Pythagoras Theorem,
$\Rightarrow AB ² = AE ² + BE ² \Rightarrow AB ² = AD ² - DE ² + BE ² . . . . from (2) \Rightarrow AB ² = (BD + DE) ² + AD ² - DE ² (∵ BE = BD + DE) \Rightarrow AB ² = BD ² + DE ² + 2 BD \times DE + AD ² - DE ² \Rightarrow AB ² = BD ² + AD ² + 2 BD \times DE \Rightarrow AB ² = (1 / 2 BC) ² + AD ² + (2 \times 1 / 2 BC \times DE) . . . . . from (1) \Rightarrow AB ² = (1 / 4 BC) ² + AD ² + BC \times DE . . . . . . (3)$

In $Δ AEC$ , applying Pythagoras Theorem,
$\Rightarrow AC ² = AE ² + EC ² \Rightarrow AC ² = AD ² - DE ² + EC ² . . . . . from (2) \Rightarrow AC ² = AD ² - DE ² + (DC - DE) ² . . . . . . (∵ EC = DC - DE) \Rightarrow AC ² = AD ² - DE ² + DC ² + DE ² - 2 DC \times DE \Rightarrow AC ² = AD ² + DC ² - 2 DC \times DE \Rightarrow AC ² = AD ² + (1 / 2 BC) ² - (2 \times 1 / 2 BC \times DE) \Rightarrow AC ² = AD ² + (1 / 4 BC) ² - BC \times DE . . . . . . (4)$

Adding $(3)$ and $(4)$ , we get
$AB ² + AC ² = 1 / 4 BC ² + AD ² + BC \times DE + AD ² + 1 / 4 BC ² - BC \times DE AB ² + AC ² = 1 / 2 BC ² + 2 AD ² 2 (AB ² + AC ²) = BC ² + 4 AD ² 2 AB ² + 2 AC ² = BC ² + 4 AD ²$

Solving for median $AD$
$\Rightarrow {AD}^{2} = \frac{2 {AB}^{2} + 2 {AC}^{2} - {BC}^{2}}{4} . \Rightarrow AD = \sqrt{\frac{2 {AB}^{2} + 2 {AC}^{2} - {BC}^{2}}{4}} \Rightarrow AD = \frac{\sqrt{2 {AB}^{2} + 2 {AC}^{2} - {BC}^{2}}}{2}$
This is the required relation.

In an acute-angled triangle, express a median in terms of its sides.

Important Questions on Triangles

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