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Tutor profile: Marilena P.

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Marilena P.
Qualified teacher with 7 years of experience
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Questions

Subject:Geometry

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Question:

A regular polygon has an interior angle that is three times the size of the exterior angle. What is the name of the regular polygon?

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Marilena P.

Let $$x$$ be the size of the exterior angle. In our question, the interior angle is three times the size of the exterior angle, therefore $$3x$$. Exterior angles are produced by extending the sides of a polygon, therefore the interior and exterior angle at any vertex of the polygon add up to 180 degrees. Interior + Exterior = $$180^{\circ}$$ $$3x + x =180^{\circ}$$ $$4x = 180^{\circ}$$ $$x = 45^{\circ}$$ We used $$x$$ to denote the size of the exterior angle, therefore the exterior angle is 45 degrees. In our question we have a $$\textbf{regular}$$ polygon. This means that the shapes is equilateral (all sides have the same length) and equiangular (its angles have the same size). Therefore, each of the exterior angles is $$45$$ degrees. The sum of the exterior angles of a regular polygon is 360 degrees. (For a nice image of this imagine the shape is getting smaller and smaller and smaller until it is just a single point where all the exterior angles meet. Angles around a point are 360 degrees.) Let $$n$$ be the number of exterior angles: $$45^{\circ} \times n = 360^{\circ}$$ $$n=8$$ Our polygon has 8 angles, therefore 8 sides. A shape with 8 sides is called an $$\textbf{\mathrm{octagon}}$$. $$\underline{\mathrm{Check}}$$: We can use the formula for the sum $$S$$ of the interior angles of a polygon $$S = (n - 2) \times 180^{\circ}$$, where $$n$$ is the number of angles In an octagon $$n=8$$, therefore: $$S=(8-2)\times 180^{\circ}$$ $$=1080^{\circ}$$ We previously found the exterior to be $$45^{\circ}$$ and the interior in our question is 3 times as big, therefore $$3\times45=135^{\circ}$$. There are 8 interior angles in an octagon, therefore the sum of the interior angles will be $$8 \times 135^{\circ} = 1080^{\circ}$$

Subject:Pre-Algebra

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Question:

If today it's Monday, what day will it be in 1000 days from today?

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Marilena P.

If today it's Monday, it will be Monday again in 7 days from today. In fact, it will Monday every 7 days (making a full circle and ending up where we started from). So it will be Monday in $$7, 14, 21, 28, 35, 42, 49, ...$$ days and so on. If $$1000$$ is a multiple of $$7$$, then in $$1000$$ days from Monday it will be Monday again. Dividing $$1000$$ by $$7$$ does not give an integer, therefore $$1000$$ is not a multiple of $$7$$. In fact: $$1000 \div 7= 142 \frac{6}{7}$$ $$7$$ goes into one thousand $$142$$ times with remainder $$6$$. $$7 \times 142= 994$$ so in $$994$$ days it will Monday again. There are $$6$$ days left after that to reach $$1000$$ and $$6$$ days after Monday it will be Sunday. So if today it's Monday, in $$1000$$ days it will be Sunday. This idea of going round in circles is the same with the $$12$$ months of the year, the $$4$$ seasons of the year, the $$24$$ hours of the day, the $$360^{\circ}$$ in a full rotation, and many others. It can also be linked to linear sequences and modular arithmetic with applications in coding and making computation faster.

Subject:Algebra

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Question:

What are the roots of $$y = x^2 + 8x + 8$$ ?

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Marilena P.

The roots are the points where the function meets the $$x$$-axis. On the $$x$$-axis the $$y$$ coordinates of all points are zero. We therefore set $$y=0$$ and solve for $$x$$: $(x^2 + 8x + 8 = 0$) We notice that $$x^2 + 8x + 8$$ does not have integer factors. We can solve by using the quadratic formula or by completing the square. $$\underline{\mathrm{Quadratic ~Formula}}$$ We can solve a quadratic equation of the form $$ax²+bx+c=0$$ by using the quadratic formula: $(x= \frac{-b±\sqrt{(b²-4ac)}}{2a}$) We substitute $$a=1, b=8, c=8$$ into the formula which gives $$x= \frac{-8±\sqrt{(8²-4(1)(8))}}{2(1)}$$ and we simplify to get $$x = -4 ±2\sqrt2$$. $$\underline{\mathrm{Completing ~the ~Square}}$$ My preferred method for this question would be to solve by completing the square. To complete the square we want to write the equation in the form of $$(x+a)^2 + c = 0$$ $$\textbf{Reminder:} ~(x+a)^2 \equiv (x+a)(x+a) \equiv x^2 + 2ax + a^2$$ Our coefficient of $$x$$ (in the term $$2ax$$ above) is $$8$$, therefore: $(2a=8\\ \,\,\,a = 4$) Squaring both sides we get $$a^2 = 16$$. We use $$a^2 = 16$$ to rewrite $$x^2 + 8x + 8 = 0$$ as: $(x^2 + 8x \textbf{ + 16 - 16} + 8 = 0$) The $$-16$$ is crucial to cancelling the effect of $$+16$$ and maintaining the original equation. Factorize $$x^2 + 8x + 16$$ to $$(x+4)^2$$ and simplify $$-16 + 8$$ to get $$-8$$. So $$x^2 + 8x \textbf{ + 16 - 16} + 8 = 0$$ can be written as: $((x+4)^2 – 8 = 0$) To get rid of $$– 8$$, we add 8 to both sides to get: $((x+4)^2 = 8$) To get rid of the power of 2, we square root both sides: $(\sqrt{(x+4)^2} = \sqrt8$) $(x+4 =\pm\sqrt8$) To get rid of $$+4$$, we subtract 4 from each side: $(x = -4 \pm\sqrt8$) Simplify $$x = -4 \pm\sqrt4 \sqrt2$$ to get: $(x = -4 \pm 2\sqrt2$) So the two roots of $$y = x^2 + 8x + 8$$ are $$x = -4 +2\sqrt2$$ and $$x = -4 -2\sqrt2$$. Since we have completed the square, we can easily find other properties of this quadratic. (i) The axis of symmetry (ii) The coordinates of the vertex We can also start to think of $$y = (x+4)^2 – 8$$ as a transformation of $$y=(x+4)^2$$ or even of $$y=x^2$$ and the fun never stops 😊 which is why I would opt for completing the square.

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