Question

geometry
unit 5
lesson 10 homework
name
date
period
minnie is designing
sunglasses. she uses a
computer drawing program
to create a diagram of the
lenses. her drawing is
shown with congruent
polygons fzdkbxln and
gmhicrwpy representing the
lenses of the sunglasses.
in polygon fzdkbxln, (overline{fz}congoverline{xb}), (overline{nl}congoverline{dr}), and (overline{dz}congoverline{kb}congoverline{lx}congoverline{nf}). the perimeter of
polygon gmhcrwpy is 56.8 millimeters (mm).
some of the side measures of gmhcrwpy are shown.
(pw = 6x - 2.4)(hc = 2(3x - 2.8))(wr = 5x + 2)

1. write an equation that can be used to determine the value of x.
2. determine the value of x.
3. determine the length of side gm.
4. determine the length of side bk.
5. determine the length of side nl.

Explanation:

Step1: Use congruent sides property

Since the polygons are congruent and \( \overline{FZ}\cong\overline{XB},\overline{NL}\cong\overline{DR},\overline{DZ}\cong\overline{KB}\cong\overline{LX}\cong\overline{NF}\), in polygon \(GMHCRWPY\), assume \(PW = HC=WR\) (because of congruent - side relations in similar - structured congruent polygons). So the equation is \(6x - 2.4=2(3x - 2.8)=5x + 2\). Using \(6x - 2.4=5x + 2\) (we can also use \(2(3x - 2.8)=5x + 2\), but \(6x - 2.4 = 5x+2\) is simpler).

Step2: Solve the equation for \(x\)

Subtract \(5x\) from both sides of \(6x - 2.4=5x + 2\):
\(6x-5x-2.4=5x - 5x+2\)
\(x-2.4 = 2\)
Add \(2.4\) to both sides: \(x=2 + 2.4=4.4\)

Step3: Find the length of \(GM\)

Since \(PW=6x - 2.4\), and \(GM = PW\) (by congruent - polygon side - equality). Substitute \(x = 4.4\) into \(6x - 2.4\):
\(GM=6\times4.4-2.4=26.4-2.4 = 24\)

Step4: Find the length of \(BK\)

Since \(WR = 5x+2\), substitute \(x = 4.4\) into \(5x + 2\): \(WR=5\times4.4+2=22 + 2=24\). And \(BK = WR\) (by congruent - polygon side - equality), so \(BK = 24\)

Step5: Find the length of \(NL\)

Since \(HC=2(3x - 2.8)\), substitute \(x = 4.4\) into \(2(3x - 2.8)\):
\(HC=2(3\times4.4-2.8)=2(13.2 - 2.8)=2\times10.4 = 20.8\). And \(NL=HC\) (by congruent - polygon side - equality), so \(NL = 20.8\)

Answer:

1. The equation is \(6x - 2.4=5x + 2\)
2. \(x = 4.4\)
3. The length of \(GM\) is \(24\) mm
4. The length of \(BK\) is \(24\) mm
5. The length of \(NL\) is \(20.8\) mm