Tag: apollonius's theorem

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Questions Related to apollonius's theorem

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 geometry similarity and right angled triangle angle theorems for a right angled triangle apollonius's theorem

Find the perimeter of an isosceles right triangle with each of its congruent as 7cm.

  1. $7\sqrt 2$ cm
  2. $14$ cm
  3. $(2+ \sqrt 2)$ cm
  4. $7(2+ \sqrt 2)$ cm
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Let the other side of triangle is x cm

Then in isosceles right angle triangle two congruent sides are 7 cm
$x^{2}=(7)^{2}+(7)^{2}$
$\Rightarrow x^{2}=49+49$
$\Rightarrow x^{2}=198$
$\Rightarrow x=7\sqrt{2}$
Then perimeter of right angle isosceles triangle =$7+7+7\sqrt{2}=14+7\sqrt{2}=7(2+\sqrt{2})$ 

So, option D is correct.

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

In a $\Delta$ $ABC, AD = 3, BC = 2, AB = 1$, find the value of $AC$. (Use Apollonius theorem).

  1. $2.35$
  2. $3.42$
  3. $4.35$
  4. $5.61$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

According to the Apollonius theorem, 
$AB^{2}+AC^{2}= 2[AD^{2}+\dfrac {BC}{2}^{2}]$
$1^{2}+AC^{2}= 2[3^{2}+\dfrac{2}{2}^{2}]$
$1+AC^{2}= 2[9 + 1]$
$AC^{2}=20 -1$
$AC^{2}= 19$
$AC = \sqrt{19}$
$AC = 4.35$

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

In a $\Delta$ $ABC, AC = 6, BC = 2, AB = 4$, find the value of $AD$. (Use Apollonius theorem).

  1. 5

  2. 4

  3. 3

  4. 2
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

According to the Apollonius theorem, 
$AB^{2}+AC^{2}= 2\left [AD^{2}+\dfrac{BC}{2}^{2}\right]$
$4^{2}+6^{2}= 2\left [AD^{2}+\dfrac{2}{2}^{2}\right]$
$16+36= 2[AD^{2} + 1]$
$2AD^{2}=52 -2$
$2AD^{2}= 50$
$AD^{2} = \dfrac{50}{2}$
$AD^{2} = 25$
$AD = 5$

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

In a $\Delta$ $ABC, AC = 8, BC = 2, AB = 6$, find the value of $AD$. (Use Apollonius theorem).

  1. 3

  2. 5

  3. 7

  4. 9
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

According to the Apollonius theorem, 
$AB^{2}+AC^{2}= 2\left [AD^{2}+\dfrac{BC}{2}^{2}\right]$
$6^{2}+8^{2}= 2\left [AD^{2}+\dfrac{2}{2}^{2}\right]$
$36+64= 2[AD^{2} + 1]$
$2AD^{2}=100 -2$
$2AD^{2}= 98$
$AD^{2} = \dfrac{98}{2}$
$AD^{2} = 49$
$AD = 7$

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

In a $\Delta$ $ABC, AC = 4, BC = 2, AB = 6$, find the value of $AD$. (Use Apollonius theorem).

  1. 2

  2. 3

  3. 4

  4. 5
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

According to the Apollonius theorem, 
$AB^{2}+AC^{2}= 2\left [AD^{2}+\dfrac{BC}{2}^{2}\right]$
$6^{2}+4^{2}= 2\left [AD^{2}+\dfrac{2}{2}^{2}\right]$
$36+16= 2[AD^{2} + 1]$
$2AD^{2}=52 -2$
$2AD^{2}= 50$
$AD^{2} = \dfrac{50}{2}$
$AD^{2} = 25$
$AD = 5$