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PSAT Math : Plane Geometry

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Example Questions

Example Question #2 : How To Find The Length Of The Diagonal Of A Hexagon

Regular hexagon ABCDEF has side length 1.

Give the length of diagonal $\overline{A E}$ .

Possible Answers:

$\frac{\sqrt{2}}{2}$

$\sqrt{2}$

$\frac{\sqrt{3}}{2}$

$\frac{3}{2}$

$\sqrt{3}$

Correct answer:

$\sqrt{3}$

Explanation:

The key is to examine $\Delta AED$ in thie figure below:

Hexagon_50

Each interior angle of a regular hexagon, including $\angle F$ , measures $120^{\circ }$ , so it can be easily deduced by way of the Isosceles Triangle Theorem that $m \angle FEA = 30^{\circ }$ . To find $m \angle AED$ we subtract $m \angle FEA = 30^{\circ }$ from $m \angle FED = 120^{\circ }$ . Thus resullting in $m \angle AED=90^{\circ}$

Also, by symmetry,

$m \angle EDA = \frac{1}{2} \cdot 120^{\circ } = 60^{\circ }$ ,

so $m \angle EAD= 30^{\circ }$ ,

and $\Delta AED$ is a $30^{\circ}-60^{\circ}-90^{\circ}$ triangle whose short leg $\overline{ED}$ has length ED = 1 .

The long leg $\overline{AE}$ of this $30^{\circ}-60^{\circ}-90^{\circ}$ triangle measures $\sqrt{3}$ times the length of short leg $\overline{ED}$ , so $AE= \sqrt{3}$ .

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Example Question #581 : Geometry

Hexagon

Note: Figure NOT drawn to scale.

The perimeter of the above hexagon is 888. Also, A-B = 10 . Evaluate .

Possible Answers:

A = 79

$A = 93\frac{4}{5}$

$A = 60\frac{1}{2}$

Insufficient information is given to answer the problem.

A = 116

Correct answer:

A = 79

Explanation:

The perimeter of the figure can be expressed in terms of the variables by adding:

$A + B + A + B + \left (2A + 2B \right ) + \left (2A + 2B \right ) = P$

Simplify and set P = 888 :

A + A + 2A + 2A + B+ B +2B + 2B= 888

6A + 6B= 888

$6\left (A + B \right )= 888$

A + B = 148

Along with A-B = 10 , we now have a system of equations to solve for by adding:

A + B = 148

$\underline{A-B =\; 10}$

$2A = \; \; \; \; \; \; 158$

A = 79

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Example Question #1 : How To Find The Length Of The Side Of A Hexagon

Hexagon

Note: Figure NOT drawn to scale.

The perimeter of the above figure is 132. What is A + B ?

Possible Answers:

$26\frac{2}{5}$

$16 \frac{1}{2}$

Correct answer:

Explanation:

The perimeter of the figure can be expressed in terms of the variables by adding:

$A + B + A + B + \left (2A + 2B \right ) + \left (2A + 2B \right ) = P$

Simplify and set P = 132 :

A + A + 2A + 2A + B+ B +2B + 2B= 132

6A + 6B= 132

$6\left (A + B \right )= 132$

A + B = 22

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Example Question #2 : Hexagons

Hexagon

Note: Figure NOT drawn to scale.

The perimeter of the above figure is 600. The ratio of to is 3:2 . Evaluate .

Possible Answers:

B = 72

B = 48

B = 100

B = 40

B = 60

Correct answer:

B = 40

Explanation:

The perimeter of the figure can be expressed in terms of the variables by adding:

$A + B + A + B + \left (2A + 2B \right ) + \left (2A + 2B \right ) = P$

Simplify and set P = 600 :

A + A + 2A + 2A + B+ B +2B + 2B= 600

6A + 6B= 600

Since the ratio of to is equivalent to 3:2 - or

$\frac{A}{B} = \frac{3}{2}$ ,

then

$A = \frac{3}{2} B$

and we can substitute as follows:

$6 \cdot \frac{3}{2}B + 6B= 600$

9B + 6B= 600

15B = 600

B = 40

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Example Question #2 : Hexagons

Possible Answers:

190

210

180

200

170

Correct answer:

190

Explanation:

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Example Question #21 : Geometry

If a triangle has 180 degrees, what is the sum of the interior angles of a regular octagon?

Possible Answers:

540

720

1260

900

1080

Correct answer:

1080

Explanation:

The sum of the interior angles of a polygon is given by 180(n - 2) where = number of sides of the polygon. An octagon has 8 sides, so the formula becomes 180(8 -2) = 1080

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Example Question #2 : How To Find An Angle In A Hexagon

In a rectangular hexagon, what is the meaure of each interior angle?

Possible Answers:

120 degrees

72 degrees

90 degrees

150 degrees

105 degrees

Correct answer:

120 degrees

Explanation:

The sum of the interior angles of a hexagon must equal 720 degrees. Because the hexagon is regular, all of the interior angles will have the same measure. A hexagon has six sides and six interior angles. Therefore, each angle measures $\frac{720}{6}=120$ .

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Example Question #1 : How To Find An Angle In A Hexagon

Hexagon

Note:Figure NOT drawn to scale.

Refer to the above figure. Evaluate .

Possible Answers:

x = 130

x = 140

x = 100

x = 110

x = 120

Correct answer:

x = 130

Explanation:

The sum of the degree measures of the angles of a (six-sided) hexagon, is

$180 ^{\circ } \times (6 - 2) = 180 ^{\circ } \times 4 = 720^{\circ }$

We can solve for in the equation

x + 92 + 164 + 90 + 137 + 107 = 720

x +590 = 720

x = 130

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Example Question #581 : Geometry

Hexagon

Note: Figure NOT drawn to scale.

Refer to the above figure. Evaluate .

Possible Answers:

x = 101

x = 111

x = 106

x = 116

x = 121

Correct answer:

x = 111

Explanation:

The sum of the degree measures of the angles of a (six-sided) hexagon, is

$180 ^{\circ } \times (6 - 2) = 180 ^{\circ } \times 4 = 720^{\circ }$

We can solve for in the equation

x + x+ 164 + 90 + 137 + 107 = 720

2x + 498 = 720

2x = 222

x=111

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Example Question #582 : Geometry

Three angles of a hexagon measure $65^{\circ}, 74^{\circ}, 89^{\circ}$ . The other three angles are congruent to one another. What is the measure of each of the latter three angles?

Possible Answers:

This hexagon cannot exist.

$164^{\circ}$

$174^{\circ}$

$48^{\circ}$

$58^{\circ}$

Correct answer:

$164^{\circ}$

Explanation:

The sum of the degree measures of the angles of a (six-sided) hexagon, is

$180 ^{\circ } \times (6 - 2) = 180 ^{\circ } \times 4 = 720^{\circ }$

Let be the common measure of the three congruent angles in question. We can solve for in the equation

x + x + x + 65+ 74 + 89 = 720

3 x + 228 = 720

3 x = 492

x = 164

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PSAT Math : Plane Geometry

Study concepts, example questions & explanations for PSAT Math

All PSAT Math Resources

Example Questions

Example Question #2 : How To Find The Length Of The Diagonal Of A Hexagon

Example Question #581 : Geometry

Example Question #1 : How To Find The Length Of The Side Of A Hexagon

Example Question #2 : Hexagons

Example Question #2 : Hexagons

Example Question #21 : Geometry

Example Question #2 : How To Find An Angle In A Hexagon

Example Question #1 : How To Find An Angle In A Hexagon

Example Question #581 : Geometry

Example Question #582 : Geometry

All PSAT Math Resources

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