Geometry - GMAT Quantitative

Card 0 of 6760

Question

Note: Figure NOT drawn to scale.

Given Regular Pentagon . What is $m \angle APN$ ?

Answer

Quadrilateral is a trapezoid, so $m \angle APN + m \angle A = 180 ^{\circ }$ .

$m \angle A = \frac{180 (5-2 )}{5} ^{\circ } = 108 ^{\circ }$ , so

$m \angle APN +108^{\circ } = 180 ^{\circ }$

$m \angle APN +108^{\circ } - 108^{\circ } = 180 ^{\circ } - 108^{\circ }$

$m \angle APN = 72^{\circ }$

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Question

The angles of a pentagon measure $x^{\circ }, \left (x + 10 \right )^{\circ }, \left (x + 10 \right )^{\circ }, \left (x + 15 \right )^{\circ }, \left (x + 25 \right )^{\circ }$ .

Evaluate .

Answer

The sum of the degree measures of the angles of a (five-sided) pentagon is $180 (5 -2 ) ^{\circ } = 540^{\circ }$ , so we can set up and solve the equation:

$x+ \left(x + 10 \right )+ \left (x + 10 \right )+ \left(x + 15 \right )+ \left(x + 25 \right ) = 540$

$5x \div 5 = 480 \div 5$

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Question

The measures of the angles of a pentagon are: $x ^{\circ },\left ( x+23 \right ) {^\circ},\left ( x+23 \right ) {^\circ},\left ( x+33 \right ) {^\circ},\left ( x+41 \right ) {^\circ}$

What is equal to?

Answer

The degree measures of the interior angles of a pentagon total $180 \left ( 5-2\right ) ^{\circ } = 540^{\circ }$ , so

$x + \left ( x+23 \right ) +\left ( x+23 \right ) +\left ( x+33 \right ) +\left ( x+41 \right ) = 540$

$5x \div 5= 420 \div 5$

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Question

What is the measure of an angle in a regular octagon?

Answer

On octagon has sides. The word regular means that all of the angles are equal. Therefore, we can use the general equation for finding the angle measurement of a regular polygon:

$\frac{(n-2)180,\textup{degrees}}{n}$ , where is the number of sides of the polygon.

$\frac{(8-2)180,\textup{degrees}}{8}=135,\textup{degrees}$ .

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Question

The perimeter of a regular hexagon is 72 centimeters. To the nearest square centimeter, what is its area?

Answer

This regular hexagon can be seen as being made up of six equilateral triangles, each formed by a side and two radii; each has sidelength $\small 72\div6=12$ centimeters. The area of one triangle is

$\small \small A = \frac{s^{2} \sqrt{3}}{4} = \frac{12^{2} \sqrt{3}}{4} \approx 62.4$

There are six such triangles, so multiply this by 6:

$\small 62.4 \cdot 6 \approx 374$

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Question

A man wants to design a room such that, looking from above, it appears as a trapezoid with a square attached (shown below). The area of the entire room is to be 100 square meters. The red line shown bisects the dotted line and has a length of 15. How many of the following answers are possible values for the length of one side of the square?

a) 5

b) 6

c) 7

d) 8

Figure is not to scale, but the trapezoidal figure will be similar in dimensions to the one shown.

Answer

Let denote the length of one side of a square. This is also the top of the trapezoid. Let denote the bottom of the trapezoid. Finally, let be the height of the trapezoid. The area of the trapezoid is then $\frac{x+y}{2}\cdot h$ while the area of the square is $x^{2}$ .

We then have the total area as 100, so: $\frac{x+y}{2}\cdot h + x^{2} = 100$

Now we know that the red line has length 15. is the region of this line that is in the trapezoid. What we notice, however, is that the remainder is precisely the length of one side of a square. So or

Rewriting the previous equation: $\frac{x+y}{2}\cdot (15-x)+ x^{2} = 100$

This is now an equation of 2 variables and we can easily cross out answers by plugging in possible values. What we find is that for $x=5\ and\ 6$ , $y=10\ and\ 8.\overline{2}$ respectively. For we get , which is too small ( must be greater than ). For we get $y=2.\overline{285714}\ \ or\ \frac{16}{7}$ .

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Question

What is the area of a regular hexagon with sidelength 10?

Answer

A regular hexagon can be seen as a composite of six equilateral triangles, each of whose sidelength is the sidelength of the hexagon:

Each of the triangles has area

$A =\frac{ s^{2}\sqrt{3}}{4}$

Substitute to get $A =\frac{ 10^{2}\sqrt{3}}{4}= \frac{ 100 \sqrt{3}}{4} = 25 \sqrt{3}$

Multiply this by 6: $25 \sqrt{3} \times 6 = 150 \sqrt{3}$ , the area of the hexagon.

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Question

What is the area of a regular octagon with sidelength 10?

Answer

The area of a regular polygon is equal to one-half the product of its apothem - the perpendicular distance from the center to a side - and its perimeter.

The perimeter of the octagon is $10 \times 8 = 80$

From the diagram below, the apothem of the octagon is .

is one half of the sidelength, or 5. can be seen to be the length of a leg of a $45^{\circ}-45^{\circ}-90^{\circ}$ triangle with hypotenuse 10, or

$\frac{10 }{ \sqrt{ 2}}= \frac{10 \cdot \sqrt{ 2}}{ \sqrt{ 2} \cdot \sqrt{ 2}} = \frac{10 \cdot \sqrt{ 2}}{ 2} = 5\sqrt{ 2}$

This makes the apothem $5 + 5\sqrt{2}$ .

The area is therefore $\frac{1}{2} \cdot 80 \cdot \left ( 5 + 5\sqrt{2} \right ) = 200 + 200 \sqrt{2}$

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Question

What is the area of the figure with vertices ?

Answer

This figure can be seen as a composite of two simple shapes: the rectangle with vertices , and the triangle with vertices .

The rectangle has length and height , so its area is the product of these dimensions, or $9 \times 10 = 90$ .

The triangle has as its base the length of the horizontal segment connecting and , which is ; its height is the vertical distance from the other vertex to this segment, which is . The area of this triangle is half the product of the base and the height, which is $\frac{1}{2} \times 10 \times 2 = 10$ .

Add the areas of the rectangle and the triangle to get the total area:

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Question

Note: Figure NOT drawn to scale

What is the area of the above figure?

Assume all angles shown in the figure are right angles.

Answer

This figure can be seen as a smaller rectangle cut out of a larger one; refer to the diagram below.

We can fill in the missing sidelengths using the fact that a rectangle has congruent opposite sides. Once this is done, we can multiply length times height of both rectangles to get the area of each, and subtract areas:

$A = 25 \cdot 10 - 15 \cdot 3 = 250 - 45 = 205$ square feet

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Question

The following picture represents a garden with a wall built around it. The garden is represented by , the gray area,; and the wall is represented by the white area.

and are both squares and the area of the garden is equal to the area of the wall.

The length of is .

Find the area of the wall.

Answer

AB's length is 7 so the area of ABCD is:

$ABCD = 7\cdot 7=49$ .

The garden area (EFGH) is equal to the wall area .

So

,

therefore

$EFGH = 0.5 \cdot ABCD$

$EFGH = 0.5 \cdot 49 = 24.5$ .

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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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Question

A pentagon with a perimeter of one mile has three congruent sides; one of the other sides is 100 feet longer than any of those three congruent sides, and the remaining side is 100 feet longer than that fourth side. What is the length of that longest side?

Answer

If each of the five congruent sides has measure , then the other two sides have measures and $\left (x + 100 \right ) +100 = x + 200$ . Add the sides to get the perimeter, which is equal to 5,280 feet, the solve for :

$5x \div 5 = 4,980 \div 5$

Each of the shortest sides is 996 feet long; the longest side is feet long.

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Question

The perimeter of a regular hexagon is one-half of a mile. Give the sidelength in inches.

Answer

One mile is 5,280 feet. The perimeter of the hexagon is one-half of this, or 2,640 feet. Since each side of a regular hexagon is congruent, the length of one side is one-sixth of this, or $2,640 \div 6 = 440$ feet.

Multiply by 12 to convert to inches: $440 \times 12 = 5,280$ inches.

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Question

The perimeter of a regular pentagon is one-fifth of a mile. Give its sidelength in feet.

Answer

One mile is 5,280 feet. The perimeter of the pentagon is one-fifth of this, or $5,280 \div 5 = 1,056$ feet. Since each side of a regular pentagon is congruent, the length of one side is one fifth-of this, or $1,056 \div 5 = 211 \frac{1}{5}$ feet.

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