Mathematics

Properties of Triangles

32 Questions

Understanding the properties of triangles is essential for solving geometry problems in competitive exams. These questions specifically address the properties of medians, right-angled triangles, and area ratios. Mastering these rules builds a strong foundation for advanced mathematics.

Triangle mediansSimilar triangles areaRight-angled trianglesApollonius theorem

Properties of Triangles Questions

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

According to Apolloneous Theorem, if $\overline AD$ is a median of $\triangle ABC$, then $AB^{2}+AC^{2}=$

  1. $AD+BD$
  2. $AD-BD$
  3. $2(AD^{2}+BD^{2})$
  4. $2(AD^{2}-BD^{2})$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$\begin{array}{l} Then,\, \, A{ B^{ 2 } }+A{ C^{ 2 } }=? &  \ AD\, \, is\, \, median &  \ \Rightarrow A{ B^{ 2 } }+A{ C^{ 2 } }=2\left| { \frac { { B{ C^{ 2 } } } }{ 4 }  } \right| +2A{ D^{ 2 } } & \left[ \begin{array}{l} BC=2BD \ or\, \, BC=DC \end{array} \right]  \ \Rightarrow A{ B^{ 2 } }+A{ C^{ 2 } }=2{ \left| { \frac { { B{ C^{  } } } }{ 2 }  } \right| ^{ 2 } }+2A{ D^{ 2 } } &  \ A{ B^{ 2 } }+A{ C^{ 2 } }=2B{ D^{ 2 } }+2A{ D^{ 2 } }=2\left( { A{ D^{ 2 } }+B{ D^{ 2 } } } \right)  &  \end{array}$