Solution to Grand Quiz Question #2

Ankita Katdare

Ankita Katdare

@abrakadabra • Oct 25, 2024
From figure we have,


[1] PR = QR
[2] QS = SR
-----------------------------------------------
find-x copy


In Figure 2, parallel to PQ, we draw SU
Because [1] and PQ || SU, we get
[3] RS =RU
[4] PS = QU
-----------------------------------------------
find-x copy 2


In Figure 3, we draw line PU.
From [4], and PQ = QP, and angle RPQ = RQP (=80),
the triangle PQS equals to triangle PQU.
So
[5] PU = QS
From [2], PU = SR
From [3], we get
[6] PU = RU
So angle RPU = 20, TPU = 10, UPQ = 60,
[7] SVU = 60
[8] PQ = QV = VP
From [5], [7] and [8], we get
[9] SU = UV = VS
-----------------------------------------------
find-x copy 3


In Figure 4, we draw UW vertical to PR.
From [6] and angle RPU = PRU, we get triangle PUW equals to triangle RUW. So
[10] PW = RW
-----------------------------------------------
find-x copy 4


Figure 5
In Figure 5, we draw RX.
From [10] and UW vertical to PR, we get
[11] angle SRX = SPX = 10
[12] RX = PX
From [11], we get
[13] angle URX = 10 = SRX
[13'] angle URX = UPX
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  • Ankita Katdare

    Ankita Katdare

    @abrakadabra • Nov 2, 2011

    find-x copy 5

    Figure 6
    In Figure 6, extend RX to PQ.
    From [13], and angle PQR = QPR = 80, we get
    line RX vertical to PQ,
    [14] line RX is the same as line RV
    From [12], [13], [14], we get triangle RTX equals to triangle PVX. So
    [15] TX = VX
    From [12], [15], we get
    [16] RV = PT
    From [13'], [16], we get triangle RVU equals to triangle PTU. So
    [17] TU = UV
    --------------------------------------------------------


    Finally
    From [9], [17], we get
    [18] TU = SU
    Because SU || PQ, so
    [19] angle RUS = RQP = 80
    From [18], [19], we get
    [20] angle STU = TSU = 50
    So angle x = angle STU - angle PTU
    = 50 - (180 - 70 - 60 - 20)
    = 50 - 30
    = 20
    --------------------------------------------------------

    This is just one of the solutions.
    The fun about geometry is that, the same problem can have different approaches to land up to a solution.

    We are expecting some logical approach (may be) different than what I mentioned above. 😀
    0
  • Ankita Katdare

    Ankita Katdare

    @abrakadabra • Nov 2, 2011

    Yesterday's question's Coins will not be credited to anyone.

    But the solution finding is still open to all.
    Please come up with better solutions.

    😀
    0
  • ISHAN TOPRE

    ISHAN TOPRE

    @ishan-nohePN • Nov 2, 2011

    AbraKaDabra
    Yesterday's question's Coins will be credited to Issue. 😀
    Thank You Madam. 😛
    By the way here is the new solution.
    If you are allowed to use a compass, actually drawing the figure will give you answer in 30 Sec. Do try it. Its fun.
    All you need to do is to draw an Isosceles triangle with any suitable base length.
    Remember that we have to play with angles. The length is immaterial. I did it. 😀
    0
  • PraveenKumar Purushothaman

    PraveenKumar Purushothaman

    @praveenkumar-66Ze92 • Nov 3, 2011

    Woah... Didn't expect it goes this much... Anyways, my solution was from Hand Drawing and CorelDraw. 😀
    0