We read your photo as:

Find the number of sides of a regular polygon if an interior angle is 3x degrees and the corresponding exterior angle is (x-20) degrees.

Mathematics

ScanSolve AI Answer

Answer

12

y=4,z=3y = 4, z = 3

P'(3, 3), Q'(6, 3), R'(6, 6), S'(3, 6)

Step-by-step solution

13 steps

  1. Use the property that the sum of an interior angle and its corresponding exterior angle of any polygon is 180∘180^\circ.

    3x∘+(x−20)∘=180∘3x^\circ + (x - 20)^\circ = 180^\circ

  2. Solve the equation for xx.

    4x−20=1804x - 20 = 180 4x=180+204x = 180 + 20 4x=2004x = 200 x=2004x = \frac{200}{4} x=50x = 50

  3. Calculate the measure of the exterior angle.

    Exterior angle =(x−20)∘=(50−20)∘=30∘= (x - 20)^\circ = (50 - 20)^\circ = 30^\circ.

  4. Find the number of sides of the polygon.

    For a regular polygon, the sum of all exterior angles is 360∘360^\circ. If nn is the number of sides, then each exterior angle is 360∘n\frac{360^\circ}{n}.

    30∘=360∘n30^\circ = \frac{360^\circ}{n} n=36030n = \frac{360}{30} n=12n = 12

    The number of sides of the polygon is 12\boxed{12}.

  5. Substitute x=−2x = -2 into the first equation.

    x+y−z=−1x + y - z = -1 −2+y−z=−1-2 + y - z = -1 y−z=−1+2y - z = -1 + 2 y−z=1(Equation 1)y - z = 1 \quad \text{(Equation 1)}

  6. Substitute x=−2x = -2 into the second equation.

    x−2y+z=−7x - 2y + z = -7 −2−2y+z=−7-2 - 2y + z = -7 −2y+z=−7+2-2y + z = -7 + 2 −2y+z=−5(Equation 2)-2y + z = -5 \quad \text{(Equation 2)}

  7. Solve the system of two equations (Equation 1 and Equation 2) for yy and zz.

    Add Equation 1 and Equation 2:

    (y−z)+(−2y+z)=1+(−5)(y - z) + (-2y + z) = 1 + (-5) y−2y−z+z=1−5y - 2y - z + z = 1 - 5 −y=−4-y = -4 y=4y = 4

  8. Substitute the value of yy into Equation 1 to find zz.

    y−z=1y - z = 1 4−z=14 - z = 1 −z=1−4-z = 1 - 4 −z=−3-z = -3 z=3z = 3

    The values are y=4,z=3\boxed{y = 4, z = 3}.

  9. Understand the enlargement transformation.

    For an enlargement with centre (0,0)(0, 0) and scale factor kk, a point (x,y)(x, y) is mapped to (kx,ky)(kx, ky).
    Given scale factor k=3k = 3.

  10. Find the image of vertex P (1, 1).

    P′=(3×1,3×1)=(3,3)P' = (3 \times 1, 3 \times 1) = (3, 3)

  11. Find the image of vertex Q (2, 1).

    Q′=(3×2,3×1)=(6,3)Q' = (3 \times 2, 3 \times 1) = (6, 3)

  12. Find the image of vertex R (2, 2).

    R′=(3×2,3×2)=(6,6)R' = (3 \times 2, 3 \times 2) = (6, 6)

  13. Find the image of vertex S (1, 2).

    S′=(3×1,3×2)=(3,6)S' = (3 \times 1, 3 \times 2) = (3, 6)

    The images of the vertices are P′(3,3),Q′(6,3),R′(6,6),S′(3,6)\boxed{P'(3, 3), Q'(6, 3), R'(6, 6), S'(3, 6)}.

Was this helpful?

Try one yourself

A new problem that tests the same skill.

Still stuck on this one?

Ask a follow-up, and the answer starts from this question.

Got a different question?Ask your own question

More Mathematics Questions

QThe size of an interior angle of a regular polygon is 3x° while its exterior is (x + 20)°. Find the…QThe interior angle of a regular polygon is 20^ exto more than three times the exterior angle. Find…QThe interior angles of a polygon are x°, 1/2x, (x+40)°, 110°, 135°, 160°. Find the value of x. 12…QAn exterior angle of a regular polygon is 25.5^ exto. Find the number of sides.QUse the formula for the exterior angle of a regular polygon.
Browse by topic:Geometry
See all Mathematics questions →
Got a syllabus? Photograph it and I'll tell you what's due and what's worth the most.
Handwritten step-by-step solution preview

Handwritten Step-by-Step Solution

Get this as a handwritten image on WhatsApp — like a tutor wrote it for you.

Try on WhatsApp