List of top Mathematics Questions on Properties of Triangles

If the angular bisector of the angle A of the triangle ABC meets its circumcircle at E and the opposite side BC at D, then $DE\cos\frac{A}{2} =$
In a triangle ABC, $a=5, b=4$ and $\tan\frac{C}{2} = \sqrt{\frac{7}{9}}$, then its inradius r =
If $a=3, b=5, c=7$ are the sides of a triangle ABC, then its circumradius is
Two ships leave a port at the same time. One of them moves in the direction of E50$^\circ$N with a speed of 8 kmph and the other moves in the direction of S20$^\circ$E with a speed of 12 kmph. Then the distance between the ships at the end of 2 hours is (in km)
Let ABC be a triangle right angled at B. If a = 13 and c = 84, then r + R =
In a triangle ABC, if $c^2 - a^2 = b(\sqrt{3}c - b)$ and $b^2 - a^2 = c(c-a)$, then $\angle ACB =$

In the following figure \(\triangle\) ABC, B-D-C and BD = 7, BC = 20, then find \(\frac{A(\triangle ABD)}{A(\triangle ABC)}\). 

In \( \triangle ABC \), ray \( BD \) bisects \( \angle ABC \), \( A - D - C \), and \( \text{seg } DE \parallel \text{side } BC \). If \( A - E - B \), then for showing \( \frac{AB}{BC} = \frac{AE}{EB} \), complete the following activity: 

\( ABCD \) is a trapezium, \( AB \parallel CD \), and the diagonals of the trapezium intersect at point \( P \). Write the answers to the following questions:
[(a)] Draw the figure using the given information.
[(b)] Write any one pair of alternate angles and opposite angles.
[(c)] Write the names of similar triangles with a test of similarity.
Segment \( PM \) is a median of \( \triangle PQR \), \( PM = 9 \), and \( PQ^2 + PR^2 = 290 \). Find \( QR \).
Prove that, "If a line parallel to a side of a triangle intersects the remaining sides in two distinct points, then the line divides the sides in the same proportion."
Find the length of the hypotenuse of a right-angled triangle if the remaining sides are \( 9 \, \text{cm} \) and \( 12 \, \text{cm} \).
Out of the dates given below, which date constitutes a Pythagorean triplet?

In $\triangle PQR$, seg $PM$ is a median. Angle bisectors of $\angle PMQ$ and $\angle PMR$ intersect sides $PQ$ and $PR$ in points $X$ and $Y$ respectively. Prove that $XY \parallel QR$. 

In $\triangle RST$, $\angle S = 90^\circ$, $\angle T = 30^\circ$, and $RT = 12$ cm, then find $RS$.
If $a$, $b$, $c$ are sides of a triangle and $a^2 + b^2 = c^2$, name the type of triangle:
The orthocentre of the triangle formed by the lines... lies in which quadrant