Tag: pythagoras theorem

Questions Related to pythagoras theorem

Multiple choice maths pythagoras theorem similarity and right angled triangle angle theorems for a right angled triangle apollonius's theorem

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

Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

For option A: 7, 24, 25. Check if right triangle: 7² + 24² = 49 + 576 = 625 = 25². Also forms valid triangle (7+24 > 25). Option A is a right triangle with hypotenuse 25 cm. The question has typo ('25 cmj') but this doesn't affect the answer.

Multiple choice maths pythagoras theorem similarity and right angled triangle angle theorems for a right angled triangle apollonius's theorem

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

Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Let the sides containing the right angle be a and b, with hypotenuse c = 20 cm. The semi-perimeter is given as 24 cm, meaning the perimeter is 48 cm and a + b + c = 48, so a + b = 28 cm. Using the Pythagorean theorem, a^2 + b^2 = 20^2 = 400. From (a + b)^2 = a^2 + b^2 + 2ab, we get 28^2 = 400 + 2ab, leading to ab = 192. Solving the quadratic equations or checking options, the sides are 12 cm and 16 cm, which are not listed in options A, B, or C. Therefore, None of these is correct.

Multiple choice maths pythagoras theorem similarity and right angled triangle angle theorems for a right angled triangle apollonius's theorem

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. $72^o$
Reveal answer Fill a bubble to check yourself
D Correct answer
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 $4\times 18^o=72^o$.

Multiple choice maths pythagoras theorem similarity and right angled triangle angle theorems for a right angled triangle apollonius's theorem

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

Reveal answer Fill a bubble to check yourself
D Correct answer
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.

Multiple choice maths pythagoras theorem similarity and right angled triangle angle theorems for a right angled triangle apollonius's theorem

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
Reveal answer Fill a bubble to check yourself
B Correct answer
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


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

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

$BC^2=260+98=358$

$BC=18.92$

Multiple choice maths pythagoras theorem similarity and right angled triangle angle theorems for a right angled triangle apollonius's theorem

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

Reveal answer Fill a bubble to check yourself
A Correct answer
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.