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ISEE Upper Level Math : Plane Geometry

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Example Questions

Example Question #11 : Hexagons

Find the perimeter of a hexagon with a side having a length of 7cm.

Possible Answers:

$49\text{cm}$

$56\text{cm}$

$42\text{cm}$

$28\text{cm}$

$35\text{cm}$

Correct answer:

$42\text{cm}$

Explanation:

A hexagon has 6 equal sides. To find the perimeter of a hexagon, we will use the following formula:

P = 6a

where a is the length of one side of the hexagon.

Now, we know one side has a length of 7cm. So, we get

$P = 6 \cdot 7\text{cm}$

$P = 42\text{cm}$

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Example Question #12 : Hexagons

What is the perimeter of a hexagon with a side length of 1?

Possible Answers:

$\frac{1}{6}$

$\textup{Cannot be determined.}$

Correct answer:

Explanation:

A hexagon has 6 similar sides.

The formula to find the perimeter of a hexagon is:

P=6s

Substitute the side into the equation.

P=6(1) = 6

The perimeter of the hexagon is .

The answer is:

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Example Question #11 : Isee Upper Level (Grades 9 12) Mathematics Achievement

If the perimeter of a septagon is equal to $5(x^{2}+1)+2x(x-4)$ , what is the length of one side? (All sides of the septagon are equal.)

Possible Answers:

$7x^{2}-4x+5$

$x^{2}-\frac{8x}{7}+\frac{5}{7}$

$x^{2}-4x+5$

$7x^{2}+1$

Correct answer:

$x^{2}-\frac{8x}{7}+\frac{5}{7}$

Explanation:

If the perimeter of a septagon (in which all sides are equal) is $5(x^{2}+1)+2x(x-4)$ , then the length of one side will be one seventh of this expression.

To find one seventh, the value must first be simplified and then divided by 7.

$5(x^{2}+1)+2x(x-4)$

$5x^{2}+5+2x^{2}-8x$

$7x^{2}-8x+5$

When this is divided by 7, the result is:

$x^{2}-\frac{8x}{7}+\frac{5}{7}$

This value is thus the length of one side of the septagon.

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Example Question #11 : Geometry

How many degrees are in an internal angle of a regular heptagon?

Possible Answers:

128.6

120

108

Correct answer:

128.6

Explanation:

The number of degrees in an internal angle of a regular polygon can be solved using the following equation where n equals the number of sides in the polygon:

$\frac{180(n-2)}{n}=\frac{180(7-2)}{7}=\frac{900}{7}=128.6$

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Example Question #12 : Geometry

What is the measure of an interior angle of a regular nonagon?

Possible Answers:

120

140

135

128.6

Correct answer:

140

Explanation:

The measure of an interior angle of a regular polygon can be determined using the following equation where n equals the number of sides:

$int=\frac{180(n-2)}{n}=\frac{180(9-2)}{9}=\frac{1260}{9}=140$

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Example Question #12 : Plane Geometry

Thingy

Note: Figure NOT drawn to scale.

Refer to the above diagram. Pentagon PENTA is regular. What is the measure of $\angle 1$ ?

Possible Answers:

$108^{\circ}$

$120^{\circ}$

$128^{\circ}$

$136^{\circ }$

$144^{\circ}$

Correct answer:

$144^{\circ}$

Explanation:

The answer can be more clearly seen by extending $\overline{PA}$ to a ray $\overrightarrow{PA}$ :

Thingy

Note that angles have been newly numbered.

$\angle 2$ and $\angle 4$ are exterior angles of a (five-sided) regular pentagon in relation to two parallel lines, so each has a measure of $\left (360 \div 5 \right )^{\circ}= 72^{\circ}$ . $\angle 3$ is a corresponding angle to $\angle 4$ , so its measure is also $72^{\circ}$ .

By angle addition,

$m \angle 1 = m\angle 2 + m\angle 3 = 72^{\circ}+ 72^{\circ} = 144^{\circ}$

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Example Question #13 : Isee Upper Level (Grades 9 12) Mathematics Achievement

Thingy

In the above figure, the seven-side polygon, or heptagon, shown is regular. What is the measure of $\angle 1$ ?

Possible Answers:

The correct answer is not given among the other responses.

$141\frac{3}{7}^{\circ }$

$128\frac{4}{7}^{\circ}^{\circ}$

$154\frac{2}{7}^{\circ}$

$102\frac{6}{7}^{\circ}^{\circ}$

Correct answer:

$154\frac{2}{7}^{\circ}$

Explanation:

The answer can be more clearly obtained by extending the top of the two parallel lines as follows:

Note that two angles have been newly labeled.

Thingy

$\angle 2$ is an interior angle of a regular heptagon and therefore has measure

$\frac{\left ( 7-2\right )180^{\circ }}{7}= 128\frac{4}{7}^{\circ }$

By the Isosceles Triangle Theorem, since the two sides of the heptagon that help form the triangle are congruent, so are the two acute angles, and

$m \angle 3 =\frac{1}{2}\left ( 180 -128\frac{4}{7} \right ) ^{\circ }= 25\frac{5}{7}^{\circ }$

$\angle 1$ is supplementary to $\angle 3$ , so

$m \angle 1 = 180^{\circ } - m \angle 3 = 180^{\circ } -25\frac{5}{7}^{\circ }= 154\frac{2}{7}^{\circ}$

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Example Question #2 : How To Find An Angle In Other Polygons

Thingy

In the above figure, the seven-side polygon, or heptagon, shown is regular. What is the measure of $\angle 1$ ?

Possible Answers:

The correct answer is not given among the other responses.

$154\frac{2}{7}^{\circ}$

$128\frac{4}{7}^{\circ}^{\circ}$

$141\frac{3}{7}^{\circ }$

$102\frac{6}{7}^{\circ}^{\circ}$

Correct answer:

$102\frac{6}{7}^{\circ}^{\circ}$

Explanation:

The answer can be more clearly seen by extending the lower right side of the heptagon to a ray, as shown:

Thingy

Note that angles have been newly numbered.

$\angle 2$ and $\angle 4$ are exterior angles of a (seven-sided) regular heptagon, so each has a measure of $\left (360 \div 7 \right )^{\circ}= 51\frac{3}{7}^{\circ}$ . $\angle 3$ is a corresponding angle to $\angle 4$ in relation to two parallel lines, so its measure is also $51\frac{3}{7}^{\circ}$ .

By angle addition,

$m \angle 1 = m\angle 2 + m\angle 3 = 51\frac{3}{7}^{\circ}^{\circ}+ 51\frac{3}{7}^{\circ}^{\circ} = 102\frac{6}{7}^{\circ}^{\circ}$

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Example Question #14 : Isee Upper Level (Grades 9 12) Mathematics Achievement

Pentagon

Note: Figure NOT drawn to scale.

In the above figure, Pentagon PENTA is regular. Give the measure of $\angle XZY$ .

Possible Answers:

$59^{\circ}$

$41 ^{\circ}$

$39 ^{\circ}$

$29 ^{\circ}$

$51 ^{\circ}$

Correct answer:

$41 ^{\circ}$

Explanation:

The sum of the degree measures of the angles of Quadrilateral XEYZ is 360, so

$m \angle XZY+ m \angle EXZ + m \angle E + m \angle EYZ = 360^{\circ}$

Each interior angle of a regular pentagon measures

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

which is therefore the measure of $\angle E$ .

It is also given that $m \angle EXZ = 90^{\circ}$ and $m \angle EYZ = 121^{\circ}$ , so substitute and solve:

$m \angle XZY+ m \angle EXZ + m \angle E + m \angle EYZ = 360^{\circ}$

$m \angle XZY+ 90^{\circ} +108^{\circ} +121^{\circ} = 360^{\circ}$

$m \angle XZY+ 319^{\circ} = 360^{\circ}$

$m \angle XZY=41 ^{\circ}$

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Example Question #3 : How To Find An Angle In Other Polygons

Pentagon

Note: Figure NOT drawn to scale.

In the above figure, Pentagon PENTA is regular. Give the measure of $\angle YZT$ .

Possible Answers:

$75^{\circ}$

$85^{\circ}$

$71^{\circ}$

$61^{\circ}$

The correct answer is not given among the other responses.

Correct answer:

$85^{\circ}$

Explanation:

The sum of the degree measures of the angles of Quadrilateral TNYZ is 360, so

$m \angle YZT + m \angle N+ m \angle T + m \angle ZYN = 360^{\circ}$ .

Each interior angle of a regular pentagon measures

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

which is therefore the measure of both $\angle N$ and $\angle T$ .

$\angle ZYN$ and $\angle ZYE$ form a linear pair, making them supplementary. Since $m \angle ZYE = 121^{\circ}$ ,

$m \angle ZYN = 180^{\circ} - m \angle ZYE = 180^{\circ} -121^{\circ} = 59^{\circ}$ .

Substitute and solve:

$m \angle YZT + 108 ^{\circ} + 108^{\circ} + 59^{\circ} = 360^{\circ}$

$m \angle YZT + 275^{\circ} = 360^{\circ}$

$m \angle YZT = 85^{\circ}$

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ISEE Upper Level Math : Plane Geometry

Study concepts, example questions & explanations for ISEE Upper Level Math

All ISEE Upper Level Math Resources

Example Questions

Example Question #11 : Hexagons

Example Question #12 : Hexagons

Example Question #11 : Isee Upper Level (Grades 9 12) Mathematics Achievement

Example Question #11 : Geometry

Example Question #12 : Geometry

Example Question #12 : Plane Geometry

Example Question #13 : Isee Upper Level (Grades 9 12) Mathematics Achievement

Example Question #2 : How To Find An Angle In Other Polygons

Example Question #14 : Isee Upper Level (Grades 9 12) Mathematics Achievement

Example Question #3 : How To Find An Angle In Other Polygons

All ISEE Upper Level Math Resources

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