Plane Geometry - SAT Math

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Question

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

Answer

The sum of the interior angles of a polygon is given by where = number of sides of the polygon. An octagon has 8 sides, so the formula becomes

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Question

Find the sum of all the inner angles in a hexagon.

Answer

To solve, simply use the formula to find the total degrees inside a polygon, where n is the number of vertices.

In this particular case, a hexagon means a shape with six sides and thus six vertices.

Thus,

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Question

The above figure shows Square . is the midpoint of $\overline{AB}$ ; is the midpoint of $\overline{CD}$ ; is the midpoint of $\overline{CN}$ . Construct $\overline{MO}$ .

. Which of the following expresses the length of $\overline{MO}$ in terms of ?

Answer

Construct $\overline{MN}$ as shown in the diagram below:

Quadrilateral is a rectangle, so opposite sides are congruent. Therefore, .

Since is the midpoint of $\overline{CD}$ ,

$CN = \frac{1}{2} CD = \frac{1}{2} t$

Since is the midpoint of $\overline{CN}$ ,

$ON = \frac{1}{2} CN = \frac{1}{2} \cdot \frac{1}{2} t = \frac{1}{4} t$ .

$\bigtriangleup MNO$ is a right triangle, so, by the Pythagorean Theorem,

$MO = \sqrt{(MN)^{2}+(NO)^{2} }$

Substituting:

$MO= \sqrt{t^{2}+\left (\frac{1}{4}t \right)^{2} }$

$MO= \sqrt{t^{2}+ \frac {1}{16} t ^{2} }$

$MO= \sqrt{ \frac {17}{16} t ^{2} }$

Apply the Product of Radicals and Quotient of Radicals Rules:

$MO= \sqrt{ \frac {17}{16} t ^{2} }$

$= \sqrt{ \frac {17}{16} } \cdot \sqrt{t^{2}}$

$= \sqrt{ \frac {17}{16} } t$

$= \frac { \sqrt{17}}{ \sqrt{16}} \cdot t$

$= \frac { \sqrt{17}}{4} t$

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Question

Two sides of a triangle have lengths 4 and 7. Which of the following represents the set of all possible lengths of the third side, x?

Answer

The set of possible lengths is: 7-4 < x < 7+4, or 3 < X < 11.

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Question

If two sides of a triangle have lengths 8 and 10, what could the length of the third side NOT be?

Answer

According to the Triangle Inequality Theorem, the sums of the lengths of any two sides of a triangle must be greater than the length of the third side. Since 10 + 8 is 18, the only length out of the answer choices that is not possible is 19.

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Question

A triangle has sides of length 8, 13, and L. Which of the following cannot equal L?

Answer

The sum of the lengths of two sides of a triangle cannot be less than the length of the third side. 8 + 4 = 12, which is less than 13.

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Question

Two sides of a triangle are 20 and 32. Which of the following CANNOT be the third side of this triangle.

Answer

Please remember the Triangle Inequality Theorem, which states that the sum of any two sides of a triangle must be greater than the third side. Therefore, the correct answer is 10 because the sum of 10 and 20 would not be greater than the third side 32.

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Question

A triangle has sides of length 5, 7, and x. Which of the following can NOT be a value of x?

Answer

The sum of the lengths of any two sides of a triangle must exceed the length of the third side; therefore, 5+7 > x, which cannot happen if x = 13.

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Question

The lengths of two sides of a triangle are 9 and 7. Which of the following could be the length of the third side?

Answer

Let us call the third side x. According to the Triangle Inequality Theorem, the sum of any two sides of a triangle must be larger than the other two sides. Thus, all of the following must be true:

x + 7 > 9

x + 9 > 7

7 + 9 > x

We can solve these three inequalities to determine the possible values of x.

x + 7 > 9

Subtract 7 from both sides.

x > 2

Now, we can look at x + 9 > 7. Subtracting 9 from both sides, we obtain

x > –2

Finally, 7 + 9 > x, which means that 16 > x.

Therefore, x must be greater than 2, greater than –2, but also less than 16. The only number that satisfies all of these requirements is 12.

The answer is 12.

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Question

The lengths of a triangle are 8, 12, and x. Which of the following inequalities shows all of the possible values of x?

Answer

According to the Triangle Inequality Theorem, the sum of any two sides of a triangle must be greater (not greater than or equal) than the remaining side. Thus, the following inequalities must all be true:

x + 8 > 12

x + 12 > 8

8 + 12 > x

Let's solve each inequality.

x + 8 > 12

Subtract 8 from both sides.

x > 4

Next, let's look at the inequality x + 12 > 8

x + 12 > 8

Subtract 12 from both sides.

x > –4

Lastly, 8 + 12 > x, which means that x < 20.

This means that x must be less than twenty, but greater than 4 and greater than –4. Since any number greater than 4 is also greater than –4, we can exclude the inequality x > –4.

To summarize, x must be greater than 4 and less than 20. We can write this as 4 < x < 20.

The answer is 4 < x < 20.

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Question

If 2 sides of the triangle are have lengths equal to 8 and 14, what is one possible length of the third side?

Answer

The sum of the lengths of 2 sides of a triangle must be greater than—but not equal to—the length of the third side. Further, the third side must be longer than the difference between the greater and the lesser of the other two sides; therefore, 20 is the only possible answer.

$4+8< 14$

$6+8= 14$

$8+14> 20$

$8+14=22$

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Question

In $\Delta ABC$ the length of AB is 15 and the length of side AC is 5. What is the least possible integer length of side BC?

Answer

Rule - the length of one side of a triangle must be greater than the differnce and less than the sum of the lengths of the other two sides.

Given lengths of two of the sides of the $\Delta ABC$ are 15 and 5. The length of the third side must be greater than 15-5 or 10 and less than 15+5 or 20.

The question asks what is the least possible integer length of BC, which would be 11

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Question

Given $\bigtriangleup ABC$ with $m \angle A > m \angle B$ and $m \angle C = 60^{\circ }$ .

Which of the following could be the correct ordering of the lengths of the sides of the triangle?

I)

II)

III)

Answer

Given two angles of unequal measure in a triangle, the side opposite the greater angle is longer than the side opposite the other angle. Therefore, we seek to find the relationship among the measures of the angles.

$m \angle C = 60^{\circ }$ , and the measures of the interior angles of a triangle total $180^{\circ }$ , so

$m \angle A + m \angle B + m \angle C = 180 ^{\circ }$

$m \angle A + m \angle B + 60 ^{\circ } = 180 ^{\circ }$

$m \angle A + m \angle B = 120 ^{\circ }$

Since $m \angle A > m \angle B$ ,

$m \angle A + m \angle A > m \angle B+ m \angle A = 120^{\circ }$

$2 m \angle A > 120^{\circ }$

$m \angle A > 60^{\circ }$ ,

and, similarly,

$m \angle B < 60^{\circ }$

Therefore,

$m \angle A > m \angle C > m \angle B$ ,

and the lengths of their opposite sides rank similarly:

.

The correct response is that only (II) can be true.

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Question

Given $\bigtriangleup ABC$ with $m \angle A > m \angle B$ and $m \angle C = 45^{\circ }$ .

Which of the following could be the correct ordering of the lengths of the sides of the triangle?

I)

II)

III)

Answer

Given two angles of unequal measure in a triangle, the side opposite the greater angle is longer than the side opposite the other angle.

If $\angle C$ were the angle of greatest measure, then $m \angle A , m \angle B < 45^{\circ }$ , and

$m \angle A + m \angle B + m \angle C < 45 ^{\circ}+ 45 ^{\circ}+ 45 ^{\circ} = 135 ^{\circ} < 180 ^{\circ}$ .

Since the measures of the angles must total $180 ^{\circ}$ , $\angle C$ cannot have the greatest measure.

$m \angle A > m \angle B$ , so we can explore two other possibilities:

$m \angle A > m \angle B > m \angle C$ , which here is possible if, for example,

$m \angle A = 85 ^{\circ }$ and $m \angle B = 50 ^{\circ }$ - since ; and,

$m \angle A > m \angle C > m \angle B$ , which here is possible if, for example,

$m \angle A = 95 ^{\circ }$ and $m \angle B = 40 ^{\circ }$ - since .

If $m \angle A > m \angle B > m \angle C$ , then ; if $m \angle A > m \angle C > m \angle B$ , then .

This makes the correct response II or III only.

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Question

Which of the given answers choices could NOT represent the sides of a triangle?

Answer

In order for three lengths to represent the sides of a triangle, they must pass the Triangle Inequality.

This means that the sum of any two sides of the triangle must exceed the length of the third side.

With the given answers, the one set of lengths that fail this test is .

$4+5 \ngtr 10$

Therefore, the lengths could not represent the sides of a triangle.

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Question

If the following shape was going to be drawn in a circle, what is the minimum radius of the circle?

Answer

IF you draw the longest diagonal across the shape, the length of it is 13.4. This means the radius must be at least 6.7. The answer is 7.

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Question

Plato High School has an unusual track in that it is shaped like a regular octagon. The track has a perimeter of two-fifths of a mile.

Boris starts at Point A and runs clockwise until he gets halfway between Point E and Point F. Which of the following responses comes closest to the number of feet he runs?

Answer

One mile comprises 5,280 feet; the perimeter of the track, two-fifths of a mile, is equal to

$\frac{2}{5} \times 5,280 = 2,112$ feet.

Each (congruent) side of the octagonal track measures one-eighth of this,

$\frac{1}{8} \times 2,112 = 264$ feet.

By running clockwise from Point A to halfway between Point E and Point F, Boris runs along four and one half sides, each of which has this length, for a total running distance of

$4 \frac{1}{2} \times 264 = 1,188$ feet.

Of the five responses, 1,200 comes closest.

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Question

Each side of the above octagonal track is 264 feet in length. Julie starts at point A and runs clockwise at a steady speed of nine miles an hour for nine minutes. When she is finished, which of the following points is closest to her?

Answer

Julie runs for nine minutes, or $\frac{9}{60} = \frac{3}{20}$ hour; she runs nine miles per hour. Setting $t = \frac{3}{20}$ and in the rate formula, we can evaluate distance in miles:

$d = rt = 9 \cdot \frac{3}{20} = \frac{27}{20 }$

Julie runs $\frac{27}{20 }$ miles, which converts to feet by multiplication by 5,280 feet per mile:

$\frac{27}{20 } \times 5,280 = 7,128$ feet.

Each side of the octagonal track measures 264 feet, so Julie runs

$7,128 \div 264 = 27$

sides of the track; this is equivalent to running the entire track three times, then three more sides. She is running clockwise, so three more sides from Point A puts her at Point D. This is the correct response.

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