Practice (47)
back to index | new
(Stewart's Theorem) Show that $$b^2m + c^2n = a(d^2 +mn)$$
Two sides of a triangle are 4 and 9; the median drawn to the third side has length 6. Find the length of the third side.
A right triangle has legs $a$ and $b$ and hypotenuse $c$. Two segments from the right angle to the hypotenuse are drawn,
dividing it into three equal parts of length $x=\frac{c}{3}$. If the segments have length $p$ and $q$, prove that $p^2 +q^2 =5x^2$.
(Apollonius’ Theorem) Let $AD$ be one median of $\triangle{ABC}$ where point $D$ lies on side $BC$. Show that the following relation holds:
$$AB^2 +AC^2 = 2\times(AD^2 +BD^2)$$
As shown, both $ABCD$ and $OPRQ$ are squares. Additionally, $O$ is the center of $ABCD$, $OP=1$, $BP=\sqrt{2}$, and $CQ=\sqrt{5}$. Find the length of $DR$.
