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SAT II Math I : Finding Sides

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Example Question #51 : 2 Dimensional Geometry

Regular Hexagon ABCDEF has perimeter 120. $\overline{AB }$ has as its midpoint; segment $\overline{MC}$ is drawn. To the nearest tenth, give the length of $\overline{MC}$ .

Possible Answers:

40.0

17.3

30.0

31.5

26.5

Correct answer:

26.5

Explanation:

The perimeter of the regular hexagon is 80, so each side measures one sixth of this, or 20. Also, since is the midpoint of $\overline{AB}$ , $AM = BM = \frac{1}{2}AB = \frac{1}{2} \cdot 20 = 10$ .

Also, each interior angle of a regular hexagon measures $120^{\circ }$ .

Below is the hexagon in question, with indicated and $\overline{MC}$ constructed; all relevant measures are marked.

Hexagon

A triangle $\bigtriangleup MBC$ is formed with MB = 10 , BC = 20 , and included angle measure $\angle B= 120 ^{\circ }$ . The length of the remaining side can be calculated using the Law of Cosines:

$c^{2} = a^{2} + b^{2} -2 ab \cos \gamma$

where and are the lengths of two sides, is the measure of their included angle, and is the length of the third side.

Setting $a = M B= 10, b= BC= 20 , \gamma= m \angle B = 120 ^{\circ }$ , and c = CM , substitute and evaluate :

$(CM)^{2} = 10^{2} + 20 ^{2} -2 (10) (20) \cos 120^{\circ }$

= 100 + 400 - 400 (-0.5)

= 500 + 200

$(CM)^{2} = 700$ ;

Taking the square root of both sides:

$CM \approx\sqrt{ 700} \approx 26.5$ ,

the correct choice.

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Example Question #491 : Sat Subject Test In Math I

Regular Pentagon ABCDE has perimeter 80. $\overline{AB }$ has as its midpoint; segment $\overline{MC}$ is drawn. To the nearest tenth, give the length of $\overline{MC}$ .

Possible Answers:

15.5

20.9

18.8

20.0

25.9

Correct answer:

20.0

Explanation:

The perimeter of the regular pentagon is 80, so each side measures one fifth of this, or 16. Also, since is the midpoint of $\overline{AB}$ , $AM = BM = \frac{1}{2}AB = \frac{1}{2} \cdot 16 = 8$ .

Also, each interior angle of a regular pentagon measures $108^{\circ }$ .

Below is the pentagon in question, with indicated and $\overline{MC}$ constructed; all relevant measures are marked.

Pentagon 2

A triangle $\bigtriangleup MBC$ is formed with MB = 8 , BC = 16 , and included angle measure $\angle B= 108^{\circ }$ . The length of the remaining side can be calculated using the law of cosines:

$c^{2} = a^{2} + b^{2} -2 ab \cos \gamma$

where and are the lengths of two sides, $\gamma$ is the measure of their included angle, and is the length of the third side.

Setting $a = MB= 8, b= BC= 16, \gamma= m \angle B = 108^{\circ }$ , and c = CM , substitute and evaluate :

$(CM)^{2} = 8^{2} + 16^{2} -2 (8) (16) \cos 108^{\circ }$

$\approx 64 + 256 -256 (-0.3090)$

$\approx 320 +79.1$

$(CM)^{2} \approx 399.1$ ;

Taking the square root of both sides:

$CM \approx\sqrt{ 399.1} \approx 20.0$ ,

the correct choice.

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Example Question #51 : Geometry

If the two legs of a right triangle are and , find the third side.

Possible Answers:

Correct answer:

Explanation:

Step 1: Recall the formula used to find the missing side(s) of a right triangle...

a^2+b^2=c^2

Step 2: Identify the legs and the hypotenuse in the formula...

a,b are the legs, and is the hypotenuse.

Step 3: Plug in the values of a and b given in the question...

a=5, b=12, c=?

(5)^2+(12)^2=c^2

$\Rightarrow 25+144=c^2$

$\Rightarrow 169=c^2$

$\Rightarrow c=\pm 13$

Step 3: A special rule about all triangles...

For any triangle, the measurements of any of the sides CANNOT BE zero.

So, the missing side is +13 , or

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Example Question #51 : Geometry

Regular Pentagon ABCDE has perimeter 80. $\overline{AB }$ and $\overline{CD}$ have and as midpoints, respectively; segment $\overline{MN}$ is drawn. Give the length of $\overline{MN}$ to the nearest tenth.

Possible Answers:

18.8

20.9

20.0

15.5

25.9

Correct answer:

20.9

Explanation:

The perimeter of the regular pentagon is 80, so each side measures one fifth of this, or 16. Since is the midpoint of $\overline{AB}$ , $AM = BM = \frac{1}{2}AB = \frac{1}{2} \cdot 16 = 8$ .

Similarly, CN = DN= 8 .

Also, each interior angle of a regular pentagon measures $108^{\circ }$ .

Below is the pentagon with the midpoints and , and with $\overline{MN}$ constructed. Note that perpendiculars have been drawn to $\overline{MN}$ from and , with feet at points and , respectively.

Pentagon 2

BCYX is a rectangle, so XY = BC = 16 .

$\sin \angle MBX = \frac{MX}{MB}$ , or $MX = MB \cdot \sin \angle MBX$

$m \angle MBX = m \angle MBC- m \angle XBC = 108 ^{\circ } - 90 ^{\circ } = 18^{\circ }$ . Substituting:

$MX = 8 \cdot \sin 18^{\circ }$

$\approx 8 \cdot 0.3090$

$\approx 2.47$

For the same reason,

$NY \approx 2.47$ .

Adding the segment lengths:

$MN = MX + XY + NY \approx 2.47+ 16 + 2.47 \approx 20.94$ ,

Rond answer to the nearest tenth.

$MN \approx 20.9$

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Example Question #61 : 2 Dimensional Geometry

Regular Pentagon ABCDE has a perimeter of eighty.

Give the length of diagonal $\overline{AC}$ to the nearest tenth.

Possible Answers:

20.9

15.5

25.9

20.0

18.8

Correct answer:

25.9

Explanation:

The perimeter of the regular pentagon is 80, so each side measures one fifth of this, or 16. Also, each interior angle of a regular pentagon measures $108^{\circ }$ .

The pentagon, along with diagonal $\overline{AC}$ , is shown below:

Pentagon 2

A triangle $\bigtriangleup ABC$ is formed with AB=BC = 16 , and included angle measure $\angle B= 108^{\circ }$ . The length of the remaining side can be calculated using the law of cosines:

$c^{2} = a^{2} + b^{2} -2 ab \cos \gamma$ ,

where and are the lengths of two sides, $\gamma$ the measure of their included angle, and the length of the side opposite that angle.

Setting $a = AB= 16, b= BC= 16, \gamma = m \angle B = 108^{\circ }$ , and c= AC , substitute and evaluate :

$(AC)^{2} = 16 ^{2} + 16^{2} -2 (16) (16) \cos 108^{\circ }$

$\approx 256 + 256 -512 (-0.3090)$

$\approx 512 + 158.21$

$(AC)^{2} \approx 670.21$

Taking the square root of both sides:

$AC \approx\sqrt{ 670.21 } \approx 25.9$ ,

the correct choice.

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Example Question #11 : How To Find The Length Of The Diagonal Of A Rhombus

ABCD is rhombus with side lengths in meters. $\overline{AB} = 13$ and $\overline{BD} = 10$ . What is the length, in meters, of $\overline{AC}$ ?

Rhombus_1

Possible Answers:

Correct answer:

Explanation:

A rhombus is a quadrilateral with four sides of equal length. Rhombuses have diagonals that bisect each other at right angles.

Rhombus_2

Thus, we can consider the right triangle AED to find the length of diagonal $\overline{AC}$ . From the given information, each of the sides of the rhombus measures meters and $\overline{BD} = 10$ .

Because the diagonals bisect each other, we know:

$\overline{AD} = 13$

$\overline{ED} = 5$

Using the Pythagorean theorem,

a^2 + b^2 = c^2

$\overline{AE}^2 + 5^2 = 13^2$

$\overline{AE}^2 = 144$

$\overline{AE} = 12$

$\overline{AC} = 24$

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SAT II Math I : Finding Sides

Study concepts, example questions & explanations for SAT II Math I

All SAT II Math I Resources

Example Questions

Example Question #51 : 2 Dimensional Geometry

Example Question #491 : Sat Subject Test In Math I

Example Question #51 : Geometry

Example Question #51 : Geometry

Example Question #61 : 2 Dimensional Geometry

Example Question #11 : How To Find The Length Of The Diagonal Of A Rhombus

All SAT II Math I Resources

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