Friday, April 6, 2018

POTW #26 - Gauss Contest May 16th

Don't forget to practice for the Gauss Contest May 16th. Past samples can be found here: http://cemc.math.uwaterloo.ca/contests/past_contests.html

And FYI that this week's POTW is great practice for our new geometry unit! 

POTW #26: new Geometry unit practice!

POTW #25 Solution:
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10 comments:

  1. Fiona BianApril 7, 2018 at 11:27 AM

    POTW:
    Info:
    - AX= AD and CY=CD
    Find:
    - Angle XDY

    The first thing I notice is that Triangle ADX is isosceles and so is Triangle DCY. So to find Angle XDY, we need to first find angles ADX and CDY (so then we can use supplementary rules to find Angle XDY). And since all the angles ina triangle equal 180, we can use algebra (yay.....) to find Triangle CDY's angles.
    (LET < BE TRIANGLE AND > BE ANGLE)

    Let x represent >CDY
    In < CDY,
    >C + >CDY+ >CYD = 180
    (Since it is isosceles...)
    >c + x + x = 180
    >c= 180-2x

    In ADX
    >A + > ADX + >AXD = 180
    y+y+2x-90= 180
    2y=270-2x
    y= 135-x

    Angle ADC would be 180 degrees (since it is straight)

    >ADX + >XDY + >CDY = 180
    y + x + >XDY= 180 (replace with variables)
    135-x + x + >XDY= 180 (replace y with "135-x")
    >XDY= 180-135 (both x's cancel each other out, flip "135" to other side to get the angle alone)
    >XDY= 45 (subtract)
    Angle XDY is 45 degrees

    ReplyDelete
  2. AnonymousApril 8, 2018 at 6:31 PM

    AX=AD
    CY=CD
    A triangle's angles always adds up to 180 degrees.
    By splitting the right triangle into different triangles, we can determine the angles of some and use those angles to calculate the rest.
    First off since there are two isosceles triangles connected to angle XDY, we can just find the angle measures of ADX and CDY.
    Angle B is 90 degrees as it is a right angle in a right triangle.
    180 - 90 = other two angle measures sum.
    90 = A + C
    Split equally, angle A and C are both equal to 45 degrees respectively.
    Since X and Y make two isosceles triangles, subtract 45 from 180 and divide by 2 too find the angle measure of angles ADX and CDY for supplementary angles.
    Each equal 67.5, which then adds up to 135 degrees.
    Using supplementary angles means that all angles add up to 180 degrees, so 180 - 135 would then be the angle measure of angle XDY.
    Angle XDY is 45 degrees.

    ReplyDelete
  3. AnonymousApril 8, 2018 at 6:41 PM

    Just realized right triangles can't have 2 same degree angles, so the actual solution would be subtract the right angle from 180, then triangle ADX is an equilateral triangle, making each angle measure 60 degrees. Angle C is then 30 degrees since angle A is 60, using the answer I found before. Angle DXY remains 45 degrees, but this time its supplementary angles are proportionate to the actual triangle.

    ReplyDelete
    Replies
    1. Fiona BianApril 14, 2018 at 11:35 AM

      Just pointing the out, saying "Angle DXY" would refer to the angle at Point X.

      Delete
  4. AnonymousApril 10, 2018 at 5:35 AM

    Grade 8 POTW

    Angle XDY has a measure of 40 degrees. I did my work on paper.

    ReplyDelete
    Replies
    1. Fiona BianApril 14, 2018 at 11:34 AM

      40 degrees? Shouldn't the angle be 45 degrees?

      Delete
  5. AnonymousApril 12, 2018 at 6:54 AM

    XDY is 45 degrees

    ReplyDelete
  6. AnonymousApril 15, 2018 at 9:26 AM

    Angle XDY is 45°. My work is done on paper.

    ReplyDelete
  7. AnonymousApril 15, 2018 at 3:13 PM

    Sry I did this kind of late.
    Grade 7/8 POTW
    Angle B = 90 degrees
    Triangle ADX and CDY are both isosceles.
    We have to find the measure of XDY.
    I will call ADX and AXD x and CDY and CYD y.
    All angles in a triangle add up to 180 degrees and all angles in a quadrilateral add up to 360 degrees (the second one will probably not be too useful.
    2x + XAD = 180
    XAD = 180 - 2x
    In the original triangle, there is a known right angle as stated before. We can now use the new information for that.
    BAC + 90 + 180 - 2x = 180
    This leads to:
    BAC + 90 - 2x = 180
    Later to:
    BAC = 2x - 90

    2y + DCY = 180
    DCY = 180 - 2x
    2y + 2x - 90 = 180
    2x + 2y = 270
    2y = 270 - 2x
    y = 135 - x
    We know ADC is a straight line making it 180 degrees.
    ADX + XDY + CDY = 180
    In other words:
    x + XDY + y
    Using substitution:
    x + XDY + 135 - x = 180
    Simplifying to:
    XDY + 135 = 180
    XDY = 180 - 135
    XDY = 45 degrees
    Angle XDY is 45 degrees.

    ReplyDelete
  8. AnonymousMay 21, 2018 at 9:16 AM

    Catch Up
    ∠XDY = 45°
    I started my analyzing the triangle(s).
    The image already told us that there was one 90°.
    I noticed that ΔAXD was an equilateral triangle, making all 3 angles of ΔAXD, 60°.

    Now we have 2 out of 3 angles of ΔABC.
    All 3 angles of a triangle must add up to 180°, therefore:
    180-60-90=30°

    We know that ΔDCY is an isosceles:
    (180-30)/2=75°

    We now have all the angles needed to find out ∠XDY.
    ∠ADX = 60°
    ∠DCY = 75°
    ∠XDY = 180-60-75= 45°

    Therefore, ∠XDY = 45°.

    ReplyDelete

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