Alternate Segment Theorem

Angles - Line in Space

Isosceles Trapeziums are Con-Cyclic

Problem : We need to prove that any isosceles trapezium is con-cyclic.

Solution :Here’s an isosceles trapezium with sides PQ and RS are parallel and PS = QR. A quadrilateral is cyclic if it’s opposite angles are supplementary (i.e. they add up to 180˚). So , we need to prove that QPS + QRS = 180˚ and PSR + PQR = 180˚.

Let us construct two perpendiculars, PU and QT, from point P and Q to segment RS.

Now, in ΔPSU and ΔQRT
PUS = QTR        by construction - both are right angles
PS = QR           Given trapezium is isosceles
PU = QT           perpendicular distance between two parallel lines
Thus, ΔPSU and ΔQRT are congruent by Right angle-Hypotenuse-Side (RHS) congruency rule
Now angles , PSU = QRT        Corresponding Parts of Congruent Triangles (CPCT)
Therefore ,angle PSR and angle QRS are equal. -------- (1)
Also, angle RQT is equal to angle SPU by CPCT. Adding right angles UPQ and TQP to the above angles, we get
RQT + TQP = SPU + UPQ
Thus, angle RQP is equal to SPQ -------- (2)
Adding equations 1 and 2 we get the following relations for angles
PSR + RQP = QRS + SPQ --------(3)
Since the sum of all the angles in a quadrilateral is 360˚,from equation (3)
PSR + RQP + QRS + SPQ = 360˚
2 (PSR + RQP) = 2 (QRS + QPS) = 360˚
PSR + RQP =QRS + QPS = 180˚
Since the opposite angles are supplementary, it can be concluded that an isosceles trapezium is a cyclic quadrilateral.

Median to Hypotenuse of a Right Triangle

Problem : Let us consider the right triangle PQR with the right angle P (Figure 1), and let PS be the median drawn from the vertex P to the hypotenuse QR. We need to find the relationship between the length of the median PS and the the length of the hypotenuse QR.

Solution :Draw a straight line passing through the midpoint S and parallel to the side PR intersecting the side PQ at the point T. (Figure 2).

The angle QPR is given as right angle. The angles QTS and QPR are equal to each other as as they are corresponding angles of the parallel lines PR and TS and the transversal PQ. Hence the angle QTS is a right angle.

As TS passes through the mid-point S and is parallel to PR , it divides the side PQ into two equal parts i.e. PT = TQ. So, the triangles PTS and QTS are right triangle triangles with equal sides PT and TQ , these triangles also have a common side TS. Hence, these triangles are congruent in as per the Side – Angle – Side (SAS) Rule. 

From this we can say that the other sides of these triangles are also equal to each other as they are the corresponding parts of the congruent triangles , thus PS = QS. Now QS is equal to half the length of the hypotenuse QR , we can say that the median PS is also equal to half the length of the hypotenuse.

Hence, we can conclude that in a right triangle , the length of median to hypotenuse is half the length of the hypotenuse.

Locus Problem - Feet of Perpendicular

Locus Problem : A ladder of length L is sliding over the floor. Find the locus of point D such that D is the foot of perpendicular dropped from point C such that OACB is a rectangle.

Solution: Let A (a,0) be the feet of the ladder and B (0,b) be the top of the ladder rest against the wall . O is the intersection point of floor and wall. OACB is a rectangle. D is the feet of the perpendicular drawn from point C onto ladder AB . As the feet of the ladder slides, the point D travels on a path called asteroid whose equation is given by x^(2/3)+y^(2/3)=L^(2/3).

In the following applet, press the play button at the lower left corner to see the various positions of the ladder as it is dragged and the movement of point D of the ladder. You can also select the ‘Show Locus’ button to see the locus and its equation.