Tag: pythagoras theorem

Questions Related to pythagoras theorem

For a right-angled triangle, two small sides are of $6$cm and $8$cm length. Length of third side will be

  1. $14$cm

  2. $9$cm

  3. $10$cm

  4. none of these


Correct Option: A

In $\Delta ABC, \, m \angle B = 90$ and $\overline{BM}$ is an altitude. If AB = 2 AM, then AC = ......

  1. 2 AM

  2. 4 AM

  3. 6 AM

  4. 8 AM


Correct Option: A

Sides of triangle are given below. Determine which of them are right triangles. In case of a right triangle, write the length of its hypotenuse.

  1. 7 cm, 24 cm, 25 cmj

  2. 3 cm, 8 cm, 6 cm

  3. 50 cm, 80 cm, 100 cm

  4. 13 cm, 12 cm, 5 cm


Correct Option: A

The hypotenuse and the semi-perimeter of right triangle are 20 cm and 24 cm, respectively. The other two sides of the triangle are :

  1. 16 cm, 15 cm

  2. 14 cm, 16 cm

  3. 20 cm, 16 cm

  4. None of these


Correct Option: A

In $\Delta PQR,\angle P$ is right angle and $\bar{PM}$ is an altitude. $PQ=\sqrt{20}$ and QM=4 then RM=____.

  1. 8

  2. 5

  3. 9

  4. 1


Correct Option: A

The length of the hypotenuse of an isosceles right triangle whose one side is $4\surd {2}\ cm$  is___________$cm$

  1. $12$

  2. $8$

  3. $8\surd {2}$

  4. $12\surd {2}$


Correct Option: A

Mark the correct alternative of the following.
In a right triangle, one of the acute angles is four times the other. Its measure is?

  1. $68^o$

  2. $84^o$

  3. $80^o$

  4. $72^o$


Correct Option: D
Explanation:

In the right-angled triangle the sum of the other two angles is $90^o$.

Let one acute angle is $x$, then the other angle is $4x$. [ Given]
Then we get,
$4x+x=90^o$
or, $5x=90^o$
or, $x=18^o$.
So the measure of that angle is $4\times 18^o=72^o$.

If the sides of a triangle are in the ratio $1\, :\, \sqrt2\, :\, 1$, then the triangle is:

  1. an equilateral triangle

  2. an isosceles triangle

  3. a right angled triangle

  4. a right angled isosceles triangle


Correct Option: D
Explanation:

Given ratio of sides of the triangle, $1 : \sqrt{2} : 1$
Let the triangle be $ABC$ and sides be
$AB = x$
$BC = x $
$AC = \sqrt{2}x$

Clearly, $AC^2 = BC^2 + AB^2$
Hence, by converse of Pythagoras theorem, $ABC$ is a right-angled isosceles triangle.

In $\triangle ABC$, AP is the median. If $AP=7$ and $AB^2+AC^2=260$, then find BC.

  1. $14$ cm

  2. $18$ cm

  3. $15$ cm

  4. $12$ cm


Correct Option: B
Explanation:

By Apollonius Theorem (states that "the sum of the squares of any two sides of any triangle equals twice the square on half the third side, together with twice the square on the median bisecting the third side".) , we have


$BC^2=AB^2+AC^2+2AP^2$

$BC^2=260+2(7)^2$

$BC^2=260+98=358$

$BC=18.92$

Find the length of median. If the sides of triangle are:
$a = 5, b = 6, c = 8$. and $m = 3, n = 2$.

  1. $\sqrt{\dfrac{206}{5}}$

  2. $\sqrt{206}$

  3. $\dfrac{\sqrt{206}}{5}$

  4. None


Correct Option: A
Explanation:
We have from Appollonius theorem,

$a(mn+p^2)=b^2m+c^2n$

$5(3\times2+p^2)=6^2\times3+8^2\times2$

$5(6+p^2)=36\times3+64\times2$

$30+5p^2=108+128$

$5p^2=236−30 \ \implies 5p^2=206$

$p^2=\dfrac{206}{5}$

$p=\sqrt{\dfrac{206}{5}}$

Option A.