The medians of a right angle triangle which are drawn from the vertices of the acute angles are 5 cm - Maths - Triangles

Question

The medians of a right angle triangle which are drawn from the
vertices of the acute angles are 5 cm and 2 √10 cm What is
the length of the hypotenuse?

I am getting the answer as √13cm , but, I am still doubtful
about the answer, so , I just wanted to confirm

Ankush Jain · Accepted Answer

Consider a right angled triangle ABC having right angle at B and A and C be the acute angle.

Given, AD and CE are the median of side BC and AB.

⇒ CD = BD and AE = BE

Let AD = 5 cm and CE = 2√10 cm

In right ΔBEC, we have

${(CE)}^{2} = {(BC)}^{2} + (BE)^{2} \Rightarrow (2 \sqrt{10})^{2} = {(BC)}^{2} + {(BE)}^{2} \Rightarrow 40 = {(BC)}^{2} + {(BE)}^{2} \Rightarrow 40 = {(2 BD)}^{2} + {(BE)}^{2} \Rightarrow {(BE)}^{2} = 40 - 4 {BD}^{2} ... (1)$

In right ΔABD, we have

${(AD)}^{2} = (AB)^{2} + (BD)^{2} {(5)}^{2} = (2 BE)^{2} + {(BD)}^{2} = 4 {BE}^{2} + (BD)^{2} \Rightarrow 25 = 4 [40 - 4 {BD}^{2}] + {BD}^{2} = 160 - 16 {BD}^{2} + {BD}^{2} \Rightarrow 25 = 160 - 15 {BD}^{2} \Rightarrow - 135 = - 15 {BD}^{2} \Rightarrow BD = \sqrt{\frac{135}{15}} \sqrt{9} = 3 cm \Rightarrow BC = 2 BD = 2 \times 3 = 6 cm$

On putting value of BD in (1), we get

${(BE)}^{2} = 40 - 4 (BD)^{2} = 40 - 4 (3)^{2} = 40 - 36 = 4 cm \Rightarrow BE = 2 cm AB = 2 BE = 2 \times 2 = 4 cm$

Again, in ΔABC, by Pythagoras theorem

${(AC)}^{2} = {(AB)}^{2} + (BC)^{2} = {(4)}^{2} + (6)^{2} = 16 + 36 \Rightarrow AC = \sqrt{52} = \sqrt{2 \times 2 \times 13} = 2 \sqrt{13} cm$

4

The medians of a right angle triangle which are drawn from the vertices of the acute angles are 5 cm and 2 √10 cm. What is the length of the hypotenuse? I am getting the answer as √13cm , but, I am still doubtful about the answer, so , I just wanted to confirm.

The medians of a right angle triangle which are drawn from the vertices of the acute angles are 5 cm and 2 √10 cm. What is the length of the hypotenuse?

I am getting the answer as √13cm , but, I am still doubtful about the answer, so , I just wanted to confirm.