Plane Geometry - SAT Math

Card 0 of 3096

Question

Angela drives 30 miles north and then 40 miles east. How far is she from where she began?

Answer

By drawing Angela’s route, we can connect her end point and her start point with a straight line and will then have a right triangle. The Pythagorean theorem can be used to solve for how far she is from the starting point: a2+b2=c2, 302+402=c2, c=50. It can also be noted that Angela’s route represents a multiple of the 3-4-5 Pythagorean triple.

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Question

Find the sum of all the angles in a pentagon.

Answer

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

In this particular case, a pentagon is a shape that has five sides and thus has five vertices.

Thus,

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Question

Dan drives 5 miles north and then 8 miles west to get to school. If he walks, he can take a direct path from his house to the school, cutting down the distance. How long is the path from Dan's house to his school?

Answer

We are really looking for the hypotenuse of a triangle that has legs of 5 miles and 8 miles.

Apply the Pythagorean Theorem:

a2 + b2 = c2

25 + 64 = c2

89 = c2

c = 9.43 miles

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Question

Given a right triangle where the two legs have lengths of 3 and 4 respectively, what is the length of the hypotenuse?

Answer

The hypotenuse can be found using Pythagorean Theorem, which is a2 + b2 = c2, so we plug in a = 3 and b = 4 to get c.

c2 =25, so c = 5

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Question

What is the hypotenuse of a right triangle with sides 5 and 8?

Answer

Because this is a right triangle, we can use the Pythagorean Theorem which says _a_2 + _b_2 = _c_2, or the squares of the two sides of a right triangle must equal the square of the hypotenuse. Here we have a = 5 and b = 8.

_a_2 + _b_2 = _c_2

52 + 82 = _c_2

25 + 64 = _c_2

89 = _c_2

c = √89

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Question

A right triangle has side lengths of 21 and 72. What is the length of the hypotenuse?

Answer

By the Pythagorean Theorem, 212 + 722 = hyp2. Then hyp2 = 5625, and the hypotenuse = 75. If you didn't know how to figure out that 752 = 5625, that's okay. Look at the answer choices. We could easily have squared them and chosen the answer choice that, when squared, equals 5625.

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Question

In the following diagram, lines and are parallel to each other. What is the value for ?

Answer

When two parallel lines are intersected by another line, the sum of the measures of the interior angles on the same side of the line is 180°. Therefore, the sum of the angle that is labeled as 100° and angle y is 180°. As a result, angle y is 80°.

Another property of two parallel lines that are intersected by a third line is that the corresponding angles are congruent. So, the measurement of angle x is equal to the measurement of angle y, which is 80°.

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Question

Two pairs of parallel lines intersect:

If A = 135o, what is 2*|B-C| = ?

Answer

By properties of parallel lines A+B = 180o, B = 45o, C = A = 135o, so 2|B-C| = 2 |45-135| = 180o

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Question

Figure not drawn to scale.

In the figure above, APB forms a straight line. If the measure of angle APC is eighty-one degrees larger than the measure of angle DPB, and the measures of angles CPD and DPB are equal, then what is the measure, in degrees, of angle CPB?

Answer

Let x equal the measure of angle DPB. Because the measure of angle APC is eighty-one degrees larger than the measure of DPB, we can represent this angle's measure as x + 81. Also, because the measure of angle CPD is equal to the measure of angle DPB, we can represent the measure of CPD as x.

Since APB is a straight line, the sum of the measures of angles DPB, APC, and CPD must all equal 180; therefore, we can write the following equation to find x:

x + (x + 81) + x = 180

Simplify by collecting the x terms.

3x + 81 = 180

Subtract 81 from both sides.

3x = 99

Divide by 3.

x = 33.

This means that the measures of angles DPB and CPD are both equal to 33 degrees. The original question asks us to find the measure of angle CPB, which is equal to the sum of the measures of angles DPB and CPD.

measure of CPB = 33 + 33 = 66.

The answer is 66.

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Question

One-half of the measure of the supplement of angle ABC is equal to the twice the measure of angle ABC. What is the measure, in degrees, of the complement of angle ABC?

Answer

Let x equal the measure of angle ABC, let y equal the measure of the supplement of angle ABC, and let z equal the measure of the complement of angle ABC.

Because x and y are supplements, the sum of their measures must equal 180. In other words, x + y = 180.

We are told that one-half of the measure of the supplement is equal to twice the measure of ABC. We could write this equation as follows:

(1/2)y = 2x.

Because x + y = 180, we can solve for y in terms of x by subtracting x from both sides. In other words, y = 180 – x. Next, we can substitute this value into the equation (1/2)y = 2x and then solve for x.

(1/2)(180-x) = 2x.

Multiply both sides by 2 to get rid of the fraction.

(180 – x) = 4x.

Add x to both sides.

180 = 5x.

Divide both sides by 5.

x = 36.

The measure of angle ABC is 36 degrees. However, the original question asks us to find the measure of the complement of ABC, which we denoted previously as z. Because the sum of the measure of an angle and the measure of its complement equals 90, we can write the following equation:

x + z = 90.

Now, we can substitute 36 as the value of x and then solve for z.

36 + z = 90.

Subtract 36 from both sides.

z = 54.

The answer is 54.

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Question

In the diagram, AB || CD. What is the value of a+b?

Answer

Refer to the following diagram while reading the explanation:

We know that angle b has to be equal to its vertical angle (the angle directly "across" the intersection). Therefore, it is 20°.

Furthermore, given the properties of parallel lines, we know that the supplementary angle to a must be 40°. Based on the rule for supplements, we know that a + 40° = 180°. Solving for a, we get a = 140°.

Therefore, a + b = 140° + 20° = 160°

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Question

The measure of the supplement of angle A is 40 degrees larger than twice the measure of the complement of angle A. What is the sum, in degrees, of the measures of the supplement and complement of angle A?

Answer

Let A represent the measure, in degrees, of angle A. By definition, the sum of the measures of A and its complement is 90 degrees. We can write the following equation to determine an expression for the measure of the complement of angle A.

A + measure of complement of A = 90

Subtract A from both sides.

measure of complement of A = 90 – A

Similarly, because the sum of the measures of angle A and its supplement is 180 degrees, we can represent the measure of the supplement of A as 180 – A.

The problem states that the measure of the supplement of A is 40 degrees larger than twice the measure of the complement of A. We can write this as 2(90-A) + 40.

Next, we must set the two expressions 180 – A and 2(90 – A) + 40 equal to one another and solve for A:

180 – A = 2(90 – A) + 40

Distribute the 2:

180 - A = 180 – 2A + 40

Add 2A to both sides:

180 + A = 180 + 40

Subtract 180 from both sides:

A = 40

Therefore the measure of angle A is 40 degrees.

The question asks us to find the sum of the measures of the supplement and complement of A. The measure of the supplement of A is 180 – A = 180 – 40 = 140 degrees. Similarly, the measure of the complement of A is 90 – 40 = 50 degrees.

The sum of these two is 140 + 50 = 190 degrees.

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Question

In rectangle ABCD, both diagonals are drawn and intersect at point E.

Let the measure of angle AEB equal x degrees.

Let the measure of angle BEC equal y degrees.

Let the measure of angle CED equal z degrees.

Find the measure of angle AED in terms of x, y, and/or z.

Answer

Intersecting lines create two pairs of vertical angles which are congruent. Therefore, we can deduce that y = measure of angle AED.

Furthermore, intersecting lines create adjacent angles that are supplementary (sum to 180 degrees). Therefore, we can deduce that x + y + z + (measure of angle AED) = 360.

Substituting the first equation into the second equation, we get

x + (measure of angle AED) + z + (measure of angle AED) = 360

2(measure of angle AED) + x + z = 360

2(measure of angle AED) = 360 – (x + z)

Divide by two and get:

measure of angle AED = 180 – 1/2(x + z)

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Question

Assume l and m are parallel horizontal lines where l is above m. A third straight line k intersects both lines l and m, creating a total of 4 different angles on each of the two intersections. The intersections of lines l and m contain angles a, b, c, d and w, x, y, z, respectively. The angle layouts start with a and w in the top left position with the other angles following in alphabetical order in the clockwise direction. What is the degree value of angle z if angle c has a value of 67 degrees?

Answer

The purpose of this question is to understand the process of using geometry to find degree values of angles.

Since l and m are parallel lines, the respective angles on both lines will have equivalent degree values. This means that a and w, b and x, and so on will be the same degree amount. A straight line is 180 degrees, so you know that angle d is degrees; and therefore angle z, which has the same value, is also 113 degrees.

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Question

If $\angle A$ measures $(40-10x)^{\circ}$ , which of the following is equivalent to the measure of the supplement of $\angle A$ ?

Answer

When the measure of an angle is added to the measure of its supplement, the result is always 180 degrees. Put differently, two angles are said to be supplementary if the sum of their measures is 180 degrees. For example, two angles whose measures are 50 degrees and 130 degrees are supplementary, because the sum of 50 and 130 degrees is 180 degrees. We can thus write the following equation:

$\dpi{100} measure\ of\ \angle A+ measure\ of\ supplement\ of\ \angle A=180$

$\dpi{100} 40-10x+ measure\ of\ supplement\ of\ \angle A=180$

Subtract 40 from both sides.

$\dpi{100} -10x+ measure\ of\ supplement\ of\ \angle A=140$

Add $\dpi{100} 10x$ to both sides.

$\dpi{100} measure\ of\ supplement\ of\ \angle A=140+10x=10x+140$

The answer is $(10x+140)^{\circ}$ .

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Question

Lines and are parallel. $\angle HFC=10^{\circ}$ , $\angle DGI=50^{\circ}$ , $\triangle EFG$ is a right triangle, and $\overline{EG}$ has a length of 10. What is the length of $\overline{EF}$

Answer

Since we know opposite angles are equal, it follows that angle $\angle AFE=10^{\circ}$ and $\angle BGE=50^{\circ}$ .

Imagine a parallel line passing through point . The imaginary line would make opposite angles with $\angle AFE$ & $\angle BGE$ , the sum of which would equal $\angle FEG$ . Therefore, $\angle FEG=60^{\circ}$ .

$\cos (60)=.5=\frac{EG}{EF}\rightarrow EF=\frac{EG}{.5}=20$

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Question

A student creates a challenge for his friend. He first draws a square, the adds the line for each of the 2 diagonals. Finally, he asks his friend to draw the circle that has the most intersections possible.

How many intersections will this circle have?

Answer

The answer to this problem is 12. This can be drawn as shown below (intersections marked in red).

We can also be sure that this is the maximal case because it is the largest answer selection. Were it not given as a multiple choice question, however, we could still be sure this was the largest. This is because no line can intersect a circle in more than 2 points. Keeping this in mind, we look at the construction of our initial shape. The square has 4 lines, and then each diagonal is an additional 2. We have thus drawn in 6 lines. The maximum number of intersections is therefore going to be twice this, or 12.

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Question

Angle $\angle ABC$ measures $20^{\circ}$

$\overrightarrow{BD}$ is the bisector of $\angle ABC$

$\overrightarrow{BE}$ is the bisector of $\angle CBD$

What is the measure of $\angle ABE$ ?

Answer

Let's begin by observing the larger angle. $\angle ABC$ is cut into two 10-degree angles by $\overrightarrow{BD}$ . This means that angles $\angle ABD$ and $\angle CBD$ equal 10 degrees. Next, we are told that $\overrightarrow{BE}$ bisects $\angle CBD$ , which creates two 5-degree angles. $\angle ABE$ consists of $\angle ABD$ , which is 10 degrees, and $\angle DBE$ , which is 5 degrees. We need to add the two angles together to solve the problem.

$\angle ABE=\angle ABD+\angle DBE$

$\angle ABE=10^{\circ}+5^{\circ}$

$\angle ABE=15^{\circ}$

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Question

The above diagram shows a pentagonal track with perimeter one third of a mile. Adrianne starts at Point A and runs clockwise until she gets halfway between Points D and E. Which of the following choices is closest to the number of feet she runs?

Answer

The perimeter of the pentagonal track is one third of a mile; one mile is equal to 5,280 feet, so the perimeter is

$\frac{1}{3} \times 5, 280 = 1,760$ feet.

Each side of the pentagon has length one fifth of its perimeter, or

$\frac{1}{5} \times 1,760 = 352$ feet.

Adrianne runs three and one half sides, or

$3 \frac{1}{2} \times 352 = 1,232$ feet.

This makes 1,200 feet the closest, and correct, choice.

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Question

Aristotle High School has an unusual track in that it is shaped like a regular pentagon. Each side of the pentagon measures 264 feet.

Benny runs at a steady speed of eight miles an hour for ten minutes, starting at point A and working his way clockwise. When he is finished, which of the following points is he closest to?

Answer

Benny runs at a rate of eight miles an hour for ten minutes, or $\frac{10}{60} = \frac{1}{6}$ hours. The distance he runs is equal to his rate multiplied by his time, so, setting $r = 8 , t = \frac{1}{6}$ in this formula:

$d = 8 \cdot \frac{1}{6} = \frac{4}{3}$ miles.

One mile comprises 5,280 feet, so this is equal to

$\frac{4}{3} \cdot 5,280 = 7,040$ feet.

Since each side of the track measures 264 feet, this means that Benny runs

$7,040 \div 264 = 26 \frac{2}{3}$ sidelengths.

This means Benny runs around the track for 25 sidelengths, which is 5 complete times, back to Point A; he then runs one more complete sidelength to Point B; and, finally, he runs $\frac{2}{3}$ of a sidelength, finishing closest to Point C.

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