Calculating the length of a diagonal of a polygon - GMAT Quantitative Reasoning

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Question

Calculate the length of the diagonal for a regular pentagon with a side length of .

Answer

A regular pentagon has five diagonals of equal length, each formed by a line going from one vertex of the pentagon to another. We can see that for one of these diagonals, an isosceles triangle is formed where the two equal side lengths between the vertices joined by the diagonal are the other two sides. If we draw a line bisecting the angle between those two sides perpendicular with the diagonal that forms the other side of the triangle, we will have two congruent right triangles whose hypotenuse is the side length, , and whose adjacent angle is half the measure of one interior angle of a pentagon. Using these two values, we can solve for the length of the opposite side, which is half of the diagonal, so we can them multiply the result by to calculate the full length of the diagonal. We start by determining the sum of the interior angles of a pentagon using the following formula, where is the number of sides of the polygon:

$180^{\circ}(n-2)=180^{\circ}(5-2)=540^{\circ}$

So to get the measure of each of the five angles in a pentagon, we divide the result by :

$\frac{540^{\circ}}{5}=108^{\circ}$

So each interior angle of a regular pentagon has a measure of $108^{\circ}$ . As explained earlier, we can find the length of half the diagonal by bisecting this angle to form two right triangles. If the hypotenuse is and the adjacent angle is half of an interior angle, or $54^{\circ}$ , then the length of the opposite side will be the hypotenuse times the sine of that angle. This only gives half of the diagonal, however, as there are two of these congruent right triangles, so we multiply the result by and we get the full length of the diagonal of a pentagon as follows:

$D=2(6\sin 54^{\circ})=9.7$

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Question

The hexagon in the above diagram is regular. If $\overline{AB}$ has length 12, which of the following expressions is equal to the length of $\overline{AD}$ ?

Answer

$\overline{AD}$ is a diameter of the regular hexagon. Examine the diagram below, which shows the hexagon with all three diameters:

Each interior angle of a hexagon measures $120^{\circ }$ , so, by symmetry, each base angle of the triangle formed is $60^{\circ }$ ; also, each central angle measures one sixth of $360^{\circ}$ , or $60^{\circ }$ . Each triangle is equilateral, so if , it follows that , and .

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Question

The octagon in the above diagram is regular. If $\overline{AB}$ has length 8, which of the following expressions is equal to the length of $\overline{AD}$ ?

Answer

Construct two other diagonals as shown.

Each of the interior angles of a regular octagon have measure $135^{\circ }$ , so it can be shown that $\bigtriangleup ABX$ is a 45-45-90 triangle. Its hypotenuse is $\overline{AB}$ , whose length is 8, so, by the 45-45-90 Triangle Theorem, the length of $\overline{AX}$ is 8 divided by $\sqrt{2}$ :

$AX= \frac{8}{\sqrt{2}}= \frac{8\cdot \sqrt{2}}{\sqrt{2}\cdot \sqrt{2}} = \frac{8 \sqrt{2}}{2} = 4 \sqrt{2}$

Likewise, $YD = 4 \sqrt{2}$ .

Since Quadrilateral is a rectangle, .

$AD = AX+XY+YD = 4\sqrt{2}+ 8 + 4 \sqrt{2}= 8+8\sqrt{2}$

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Question

Note: Figure NOT drawn to scale.

Which of the following statements is true of the length of $\overline{AC}$ ?

Answer

By dividing the figure into rectangles and taking advantage of the fact that opposite sides of rectangles are congruent, we have the following sidelengths:

$\overline{AC}$ is the hypotenuse of a triangle with legs of lengths 8 and 16, so its length can be calculated using the Pythagorean Theorem:

$AC = \sqrt{(AB)^{2}+(BC)^{2}} = \sqrt{16^{2}+8^{2}} = \sqrt{256+64} = \sqrt{320}$

The question can now be answered by noting that $17^{2} = 289$ and $18^{2}= 324$ .

$17^{2}< 320 < 18^{2}$ ,

so $\sqrt{320}$ falls between 17 and 18.

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Question

Note: Figure NOT drawn to scale.

Refer to the above diagram. What is the length of $\overline{AD}$ in terms of ?

Answer

Extend sides $\overline{CD}$ and $\overline{ED}$ as shown to divide the polygon into three rectangles:

Taking advantage of the fact that opposite sides of a rectangle are congruent, we can find and :

$\bigtriangleup DXA$ is right, so by the Pythagorean Theorem,

$AD = \sqrt{(AX)^{2}+(DX)^{2}}$

$=\sqrt{\left ( n+4\right )^{2}+n^{2}}$

$=\sqrt{\left ( n^{2}+8n+16\right )+n^{2}}$

$=\sqrt{2 n^{2}+8n+16 }$

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Question

Each side of convex Pentagon has length 12. Also, $m \angle A = m \angle B = m \angle D = m\angle E = 105 ^{\circ }$ .

Construct diagonal $\overline{BD}$ . What is its length?

Answer

The measures of the interior angles of a convex pentagon total

$180 ^{\circ } \times (5-2) = 540 ^{\circ }$ ,

so

$m \angle C = 540 ^{\circ } - (m \angle A+m \angle B + m\angle D + m \angle E)$

$= 540 ^{\circ } - (105 ^{\circ }+105 ^{\circ }+105 ^{\circ }+105 ^{\circ })$

$= 540 ^{\circ } - 420 ^{\circ }$

$= 120^{\circ }$

The pentagon referenced is the one below. Note that the diagonal $\overline{BD}$ , along with congruent sides $\overline{BC}$ and $\overline{CD}$ , form an isosceles triangle $\bigtriangleup BCD$ .

Now construct the altitude from to $\overline{BD}$ :

$\overline{CM}$ bisects $\angle BCD$ and $\overline{BD}$ to form two 30-60-90 triangles. Therefore, $CM = \frac{1}{2} BC = \frac{1}{2} \cdot 12 = 6$ ,

and $BM = CM \cdot \sqrt{3}= 6\sqrt{3}$ .

$BD = 2 \cdot BM = 2 \cdot 6 \sqrt{3}= 12 \sqrt{3}$

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