Related Topics:
Geometry
These are some of the topics in Geometry that you will need to know for the New SAT Math Test (March 2016 and beyond).
Lines and Angles: Lengths and midpoints, Vertical angles, Straight angles and the sum of angles about a point, Properties of parallel lines and angles formed by a transversal, Properties of perpendicular lines.
Triangles and other polygons: Right triangles, Pythagorean Triples and the Pythagorean Theorem, Properties of equilateral and isosceles triangles, Properties of 30°60°90° triangles and 45°45°90° triangles, Congruent triangles and other congruent figures, Similar triangles and other similar figures, The triangle inequality, Quadrilaterals, Regular polygons.
Circles: Radius, diameter and circumference, Measure of central angles and inscribed angles, Arc Length and area of sectors, Tangents and chords.
Solids: Area, Surface Area and Volume
New SAT Practice: Lines, Angles, Triangles
Examples:
1. The area of triangle DEF is 32√3 square units. What is the length of DF?
2. In the figure, what is the area of triangle QRS?
3. If AD = 2CD, BD = BC = 6, what is the length of side AB?
4. A triangle is graphed on a standard coordinate plane. What is the perimeter of the triangle if it has vertices (1,4), (1,7), and (4,4)?
5. What is the area, in square units, of an isosceles right triangle with hypotenuse of 2?
New SAT Practice: Lines, Angles, Triangles (Hard)
Examples:
1. Figure ABCD is a parallelogram. What is the product of xy?
2. Each of the quadrilaterals is a square. The area of the smallest square, square 1, is 16 square units, the area of the medium square, square 2, is 48 square units. What is the area, in square units, of the largest square?
3. Find the length of BC.
4. Given that the sides of a right triangle are x, x  2 and 2x  2, what is the value of x?
New SAT Practice: Similarity, Congruence, Proofs (Easy)
Examples:
1. Confirmation of which of the following facts would be sufficient to prove that the two triangles are congruent?
2. While working on a geometry problem, Raul determines that the angles of one triangle are congruent to the corresponding angles of another triangle. Which of the following is a valid deduction that Raul can make?
A. The two triangles are congruent but not necessarily similar.
B. The two triangles are similar, but not necessarily congruent.
C. The two triangle are both similar and congruent.
D. The two triangles are neither similar nor congruent.
New SAT Practice: Similarity, Congruence, Proofs (Medium)
Examples:
1. AC and BD are diameters of circle E. Which triangle congruence theorem can be used to prove triangle AEB is congruent to triangle DEC?
2. If the right triangles are similar triangles, what is the length of the shorter leg of the larger triangle?
3. A triangle with side lengths 5, 12, 15 centimeters is similar to another triangle. The longest side of the other triangle has length 24 centimeters. What is the perimeter, in centimeters, of the larger triangle?
4. Two triangles are graphed on the coordinate plane. Triangle MNP has vertices M(4,2), N(4,6), P(6,2). Triangle QRS has vertices Q(5,1), R(5,5), S(4,5). Which of the following statements is true?
A. Triangle MNP is congruent to triangle QRS.
B. Triangle MNP is similar to triangle QRS.
C. Triangle MNP is similar to triangle RSQ.
D. The triangles are neither congruent nor similar.
5. JK = KL, JL = 26. The ratio of MV to MP is 8:5. Find the length of JM?
New SAT Practice: Similarity, Congruence, Proofs (Hard)
Examples:
1. Triangle CAT is an isosceles triangle with vertices (2,1), (6,1), (4,7). Triangle DOG is similar to CAT, and two of its vertices are (3,1) and (5,1). If the third vertex has a ycoordinate that is less than 1, what are the coordinates of the third vertex?
2. UY and WX are parallel. UX intersects WY at V. What is the length of WY?
3. A scientist looking at a sample of infected tissue through a microscope wants to find the length x, in microns, across a damaged blood cell. The lengths represented by AB, EB, BD and CD are determined to be 26 microns, 22 microns, 11 microns, and 12 microns, respectively. Given that the measure of AEB is equal to the measure of CBD, what is the value of x?
The following materials relate to the SAT before March 2016.
For updated SAT materials, please see:
Try out the following Geometry Questions to help you prepare for the SAT, ACT or other similar examinations.
(Note: You may need to enable JavaScript on your browser for the hints and solutions to show properly.)
Question 6
John has a square piece of paper with sides 4 inches each. He rolled up the paper to form a cylinder. What is the volume of the cylinder?
(A)


(B) 

(C) 
16 
(D) 
4π 
(E) 
16π 
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Question 7
The square ABCD touches the circle at 4 points. The length of the side of the square is 2 cm. Find the area of the shaded region.
(A)
(B)
(C)
(D)
(E) 
π – 4
2π – 4
3π – 4
4π – 4
5π – 4 
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Question 8
Note: Figures not drawn to scale
In the figure above, if x < 90° then which of the following must be true?
(A)

l ∥ m 
(B) 
l ⊥ m 
(C) 
y < 90° 
(D) 
y = 90° 
(E) 
y > 90° 
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Solution:
Step 1: x and y are supplementary angles. So if x < 90° then y > 90°
Beware! Although the figure seems to show that l is parallel to m, it is not
stated in the question.
Answer: (E) y > 90°
Question 9
Note: Figures not drawn to scale
In the figure above, _{} and _{} and AE = EF . What is the value of x?
(A)
(B)
(C)
(D)
(E) 
40
45
50
55
60

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Solution:
Step 1: We can find the supplementary angle of 120°, which gives us 60°.
Step 2: is parallel to because they are both perpendicular to . This means that since they are corresponding angles.
Step 3: Triangle ABE and triangle ACF are similar triangles , since and (AA rule). Given that AE = EF, we can conclude that AB = BC
Step 4: Triangle ABE and triangle CBE are congruent triangles (SAS rule). So, x = angle AEB = 60°
So, x = 60
Answer: (E) 60
Question 10
In the figure above, if ∠AOB = 40° and the length of arc AB is 4π, what is the area of the sector AOB?
(A)
(B)
(C)
(D)
(E) 
4π
16π
36π
128π
324π

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Example:
Which segment is congruent to BE?
Example:
Which line has a slope of 3?
Example:
AD is parallel to BC. AC = 13, BC = 5, BD = 15. What is the length of AD?
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