4GMAT Diagnostic Test Q8 - Problem Solving : Simple and Compound Interest
4GMAT Diagnostic Test Q6 - Problem Solving - Geometry, Triangles
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GMAT QUANTITATIVE REASONING GEOMETRY TRIANGLE BASICS PROBLEM SOLVING Diagnostic Test
Transcript of 4GMAT Diagnostic Test Q6 - Problem Solving - Geometry, Triangles
GMAT QUANTITATIVE REASONING
GEOMETRY
TRIANGLE BASICS
PROBLEM SOLVING
Diagnostic Test
Question
In triangle ABC, which of the following could be the perimeter of the triangle if side AB measures 5 units and side BC measures 7 units?
I. 15 II. 25 III. 17
A. I onlyB. II onlyC. I and II onlyD. I and III onlyE. I, II, and III
Part 1
Basic Property of Triangles
Triangle Property
Property relating to sides of a triangle
Triangle Property
Property relating to sides of a triangle
Sum of any two sides should be greater than the third side.
Triangle Property
Property relating to sides of a triangle
Sum of any two sides should be greater than the third side.
AB + BC > AC; AB + AC > BC; BC + AC > AB
Triangle Property
Property relating to sides of a triangleFor practical reasons one can rewrite the property as
Sum of two smaller sides > longer side
Sum of any two sides should be greater than the third side.
AB + BC > AC; AB + AC > BC; BC + AC > AB
Part 2
Using the property to solve the question
Which of the following could be the perimeter of the triangle?Side AB measures 5 units and side BC measures 7 units.
Options given: I. 15 II. 25 III. 17
Step 1: Compute the range for side AC
Which of the following could be the perimeter of the triangle?Side AB measures 5 units and side BC measures 7 units.
Options given: I. 15 II. 25 III. 17
What is the least value of side AC?
Step 1: Compute the range for side AC
Which of the following could be the perimeter of the triangle?Side AB measures 5 units and side BC measures 7 units.
Options given: I. 15 II. 25 III. 17
What is the least value of side AC?
Step 1: Compute the range for side AC
While computing least value, AC will be one of the smaller sides.
Which of the following could be the perimeter of the triangle?Side AB measures 5 units and side BC measures 7 units.
Options given: I. 15 II. 25 III. 17
What is the least value of side AC?
Step 1: Compute the range for side AC
While computing least value, AC will be one of the smaller sides.
BC is the longest of the 3 sides.
Which of the following could be the perimeter of the triangle?Side AB measures 5 units and side BC measures 7 units.
Options given: I. 15 II. 25 III. 17
What is the least value of side AC?
Step 1: Compute the range for side AC
While computing least value, AC will be one of the smaller sides.
BC is the longest of the 3 sides.
AB + AC > BC
Which of the following could be the perimeter of the triangle?Side AB measures 5 units and side BC measures 7 units.
Options given: I. 15 II. 25 III. 17
What is the least value of side AC?
Step 1: Compute the range for side AC
While computing least value, AC will be one of the smaller sides.
BC is the longest of the 3 sides.
AB + AC > BC
i.e., 5 + AC > 7 or AC > 2
Which of the following could be the perimeter of the triangle?Side AB measures 5 units and side BC measures 7 units.
Options given: I. 15 II. 25 III. 17
What is the least value of side AC?
Step 1: Compute the range for side AC
While computing least value, AC will be one of the smaller sides.
BC is the longest of the 3 sides.
AB + AC > BC
i.e., 5 + AC > 7 or AC > 2
The least value for AC is a delta more than 2.
Which of the following could be the perimeter of the triangle?Side AB measures 5 units and side BC measures 7 units.
Options given: I. 15 II. 25 III. 17
What is the least value of side AC? What is the largest value of side AC?
Step 1: Compute the range for side AC
While computing least value, AC will be one of the smaller sides.
BC is the longest of the 3 sides.
AB + AC > BC
i.e., 5 + AC > 7 or AC > 2
The least value for AC is a delta more than 2.
Which of the following could be the perimeter of the triangle?Side AB measures 5 units and side BC measures 7 units.
Options given: I. 15 II. 25 III. 17
What is the least value of side AC? What is the largest value of side AC?
Step 1: Compute the range for side AC
While computing least value, AC will be one of the smaller sides.
BC is the longest of the 3 sides.
AB + AC > BC
i.e., 5 + AC > 7 or AC > 2
The least value for AC is a delta more than 2.
While computing largest value, AC will be the longest side.
Which of the following could be the perimeter of the triangle?Side AB measures 5 units and side BC measures 7 units.
Options given: I. 15 II. 25 III. 17
What is the least value of side AC? What is the largest value of side AC?
Step 1: Compute the range for side AC
While computing least value, AC will be one of the smaller sides.
BC is the longest of the 3 sides.
AB + AC > BC
i.e., 5 + AC > 7 or AC > 2
The least value for AC is a delta more than 2.
While computing largest value, AC will be the longest side.
AB + BC > AC
Which of the following could be the perimeter of the triangle?Side AB measures 5 units and side BC measures 7 units.
Options given: I. 15 II. 25 III. 17
What is the least value of side AC? What is the largest value of side AC?
Step 1: Compute the range for side AC
While computing least value, AC will be one of the smaller sides.
BC is the longest of the 3 sides.
AB + AC > BC
i.e., 5 + AC > 7 or AC > 2
The least value for AC is a delta more than 2.
While computing largest value, AC will be the longest side.
AB + BC > AC
i.e., 5 + 7 > AC or AC < 12
Which of the following could be the perimeter of the triangle?Side AB measures 5 units and side BC measures 7 units.
Options given: I. 15 II. 25 III. 17
What is the least value of side AC? What is the largest value of side AC?
Step 1: Compute the range for side AC
While computing least value, AC will be one of the smaller sides.
BC is the longest of the 3 sides.
AB + AC > BC
i.e., 5 + AC > 7 or AC > 2
The least value for AC is a delta more than 2.
While computing largest value, AC will be the longest side.
AB + BC > AC
i.e., 5 + 7 > AC or AC < 12
The largest value for AC is a delta less than 12.
Which of the following could be the perimeter of the triangle?Side AB measures 5 units and side BC measures 7 units.
Options given: I. 15 II. 25 III. 17
What is the least value of side AC? What is the largest value of side AC?
Step 1: Compute the range for side AC
While computing least value, AC will be one of the smaller sides.
BC is the longest of the 3 sides.
AB + AC > BC
i.e., 5 + AC > 7 or AC > 2
The least value for AC is a delta more than 2.
While computing largest value, AC will be the longest side.
AB + BC > AC
i.e., 5 + 7 > AC or AC < 12
The largest value for AC is a delta less than 12.
2 < AC < 12
Side AB measures 5 units and side BC measures 7 units. 2 < side AC < 12
Options given: I. 15 II. 25 III. 17
Which of the following could be the perimeter of the triangle?
Step 2: Compute the range for the perimeter
Side AB measures 5 units and side BC measures 7 units. 2 < side AC < 12
Options given: I. 15 II. 25 III. 17
What is the least value of the perimeter?
Which of the following could be the perimeter of the triangle?
Step 2: Compute the range for the perimeter
Side AB measures 5 units and side BC measures 7 units. 2 < side AC < 12
Options given: I. 15 II. 25 III. 17
What is the least value of the perimeter?
Which of the following could be the perimeter of the triangle?
Step 2: Compute the range for the perimeter
To compute least value of perimeter use lower bound of AC.
Side AB measures 5 units and side BC measures 7 units. 2 < side AC < 12
Options given: I. 15 II. 25 III. 17
What is the least value of the perimeter?
Which of the following could be the perimeter of the triangle?
Step 2: Compute the range for the perimeter
To compute least value of perimeter use lower bound of AC.
Perimeter (P) is AB + BC + AC
Side AB measures 5 units and side BC measures 7 units. 2 < side AC < 12
Options given: I. 15 II. 25 III. 17
What is the least value of the perimeter?
Which of the following could be the perimeter of the triangle?
Step 2: Compute the range for the perimeter
To compute least value of perimeter use lower bound of AC.
Perimeter (P) is AB + BC + AC
i.e., P = 5 + 7 + (value greater than 2)
Side AB measures 5 units and side BC measures 7 units. 2 < side AC < 12
Options given: I. 15 II. 25 III. 17
What is the least value of the perimeter?
Which of the following could be the perimeter of the triangle?
Step 2: Compute the range for the perimeter
To compute least value of perimeter use lower bound of AC.
Perimeter (P) is AB + BC + AC
i.e., P = 5 + 7 + (value greater than 2)
i.e., P > 14
Side AB measures 5 units and side BC measures 7 units. 2 < side AC < 12
Options given: I. 15 II. 25 III. 17
What is the least value of the perimeter? What is the largest value of the perimeter?
Which of the following could be the perimeter of the triangle?
Step 2: Compute the range for the perimeter
To compute least value of perimeter use lower bound of AC.
Perimeter (P) is AB + BC + AC
i.e., P = 5 + 7 + (value greater than 2)
i.e., P > 14
Side AB measures 5 units and side BC measures 7 units. 2 < side AC < 12
Options given: I. 15 II. 25 III. 17
What is the least value of the perimeter? What is the largest value of the perimeter?
Which of the following could be the perimeter of the triangle?
Step 2: Compute the range for the perimeter
To compute least value of perimeter use lower bound of AC.
Perimeter (P) is AB + BC + AC
i.e., P = 5 + 7 + (value greater than 2)
i.e., P > 14
To compute largest value of perimeter use upper bound of AC.
Side AB measures 5 units and side BC measures 7 units. 2 < side AC < 12
Options given: I. 15 II. 25 III. 17
What is the least value of the perimeter? What is the largest value of the perimeter?
Which of the following could be the perimeter of the triangle?
Step 2: Compute the range for the perimeter
To compute least value of perimeter use lower bound of AC.
Perimeter (P) is AB + BC + AC
i.e., P = 5 + 7 + (value greater than 2)
i.e., P > 14
To compute largest value of perimeter use upper bound of AC.
i.e., P = 5 + 7 + (value lesser than 12)
Side AB measures 5 units and side BC measures 7 units. 2 < side AC < 12
Options given: I. 15 II. 25 III. 17
What is the least value of the perimeter? What is the largest value of the perimeter?
Which of the following could be the perimeter of the triangle?
Step 2: Compute the range for the perimeter
To compute least value of perimeter use lower bound of AC.
Perimeter (P) is AB + BC + AC
i.e., P = 5 + 7 + (value greater than 2)
i.e., P > 14
To compute largest value of perimeter use upper bound of AC.
i.e., P = 5 + 7 + (value lesser than 12)
i.e., P < 24
Side AB measures 5 units and side BC measures 7 units. 2 < side AC < 12
Options given: I. 15 II. 25 III. 17
What is the least value of the perimeter? What is the largest value of the perimeter?
Which of the following could be the perimeter of the triangle?
Step 2: Compute the range for the perimeter
To compute least value of perimeter use lower bound of AC.
Perimeter (P) is AB + BC + AC
i.e., P = 5 + 7 + (value greater than 2)
i.e., P > 14
To compute largest value of perimeter use upper bound of AC.
i.e., P = 5 + 7 + (value lesser than 12)
i.e., P < 24
14 < P < 24
Side AB measures 5 units and side BC measures 7 units. 14 < Perimeter (P) < 24
Options given: I. 15 II. 25 III. 17
Which of the following could be the perimeter of the triangle?
Step 3: Check which options fall within the range
Side AB measures 5 units and side BC measures 7 units. 14 < Perimeter (P) < 24
Options given: I. 15 II. 25 III. 17
Which of the following could be the perimeter of the triangle?
Step 3: Check which options fall within the range
14 < Perimeter < 24
Side AB measures 5 units and side BC measures 7 units. 14 < Perimeter (P) < 24
Options given: I. 15 II. 25 III. 17
I. 15
Which of the following could be the perimeter of the triangle?
Step 3: Check which options fall within the range
14 < Perimeter < 24
Side AB measures 5 units and side BC measures 7 units. 14 < Perimeter (P) < 24
Options given: I. 15 II. 25 III. 17
II. 25I. 15
Which of the following could be the perimeter of the triangle?
Step 3: Check which options fall within the range
14 < Perimeter < 24
Side AB measures 5 units and side BC measures 7 units. 14 < Perimeter (P) < 24
Options given: I. 15 II. 25 III. 17
III. 17II. 25 I. 15
Which of the following could be the perimeter of the triangle?
Step 3: Check which options fall within the range
14 < Perimeter < 24
Side AB measures 5 units and side BC measures 7 units. 14 < Perimeter (P) < 24
Options given: I. 15 II. 25 III. 17
III. 17II. 25
Options I and III
I. 15
Which of the following could be the perimeter of the triangle?
Step 3: Check which options fall within the range
14 < Perimeter < 24
Side AB measures 5 units and side BC measures 7 units. 14 < Perimeter (P) < 24
Options given: I. 15 II. 25 III. 17
III. 17II. 25
Correct Answer choice D.
Options I and III
I. 15
Which of the following could be the perimeter of the triangle?
Step 3: Check which options fall within the range
14 < Perimeter < 24
Part 2 – Alternative Method
Use answer choices. Smarter method.
Which of the following could be the perimeter of the triangle?Side AB measures 5 units and side BC measures 7 units.
Options given: I. 15 II. 25 III. 17
Approach: Compute AC for each option and see if it satisfies the basic property relating to
sides of a triangle
Which of the following could be the perimeter of the triangle?Side AB measures 5 units and side BC measures 7 units.
Options given: I. 15 II. 25 III. 17
Approach: Compute AC for each option and see if it satisfies the basic property relating to
sides of a triangle
Sum of two smaller sides should be greater than the longer side.
I: Perimeter 15
Which of the following could be the perimeter of the triangle?Side AB measures 5 units and side BC measures 7 units.
Options given: I. 15 II. 25 III. 17
Approach: Compute AC for each option and see if it satisfies the basic property relating to
sides of a triangle
Sum of two smaller sides should be greater than the longer side.
I: Perimeter 15
Which of the following could be the perimeter of the triangle?Side AB measures 5 units and side BC measures 7 units.
Options given: I. 15 II. 25 III. 17
Approach: Compute AC for each option and see if it satisfies the basic property relating to
sides of a triangle
Sum of two smaller sides should be greater than the longer side.
5 + 7 + AC = 15
I: Perimeter 15
Which of the following could be the perimeter of the triangle?Side AB measures 5 units and side BC measures 7 units.
Options given: I. 15 II. 25 III. 17
Approach: Compute AC for each option and see if it satisfies the basic property relating to
sides of a triangle
Sum of two smaller sides should be greater than the longer side.
5 + 7 + AC = 15
Or AC = 15 – 12 = 3
I: Perimeter 15
Which of the following could be the perimeter of the triangle?Side AB measures 5 units and side BC measures 7 units.
Options given: I. 15 II. 25 III. 17
Approach: Compute AC for each option and see if it satisfies the basic property relating to
sides of a triangle
Sum of two smaller sides should be greater than the longer side.
5 + 7 + AC = 15
Or AC = 15 – 12 = 3
Is 5 + 3 > 7?
I: Perimeter 15
Which of the following could be the perimeter of the triangle?Side AB measures 5 units and side BC measures 7 units.
Options given: I. 15 II. 25 III. 17
Approach: Compute AC for each option and see if it satisfies the basic property relating to
sides of a triangle
Sum of two smaller sides should be greater than the longer side.
5 + 7 + AC = 15
Or AC = 15 – 12 = 3
Is 5 + 3 > 7? Yes.
I: Perimeter 15 II: Perimeter 25
Which of the following could be the perimeter of the triangle?Side AB measures 5 units and side BC measures 7 units.
Options given: I. 15 II. 25 III. 17
Approach: Compute AC for each option and see if it satisfies the basic property relating to
sides of a triangle
Sum of two smaller sides should be greater than the longer side.
5 + 7 + AC = 15
Or AC = 15 – 12 = 3
Is 5 + 3 > 7? Yes.
I: Perimeter 15 II: Perimeter 25
Which of the following could be the perimeter of the triangle?Side AB measures 5 units and side BC measures 7 units.
Options given: I. 15 II. 25 III. 17
Approach: Compute AC for each option and see if it satisfies the basic property relating to
sides of a triangle
Sum of two smaller sides should be greater than the longer side.
5 + 7 + AC = 15
Or AC = 15 – 12 = 3
Is 5 + 3 > 7? Yes.
5 + 7 + AC = 25
I: Perimeter 15 II: Perimeter 25
Which of the following could be the perimeter of the triangle?Side AB measures 5 units and side BC measures 7 units.
Options given: I. 15 II. 25 III. 17
Approach: Compute AC for each option and see if it satisfies the basic property relating to
sides of a triangle
Sum of two smaller sides should be greater than the longer side.
5 + 7 + AC = 15
Or AC = 15 – 12 = 3
Is 5 + 3 > 7? Yes.
5 + 7 + AC = 25
Or AC = 25 – 12 = 13
I: Perimeter 15 II: Perimeter 25
Which of the following could be the perimeter of the triangle?Side AB measures 5 units and side BC measures 7 units.
Options given: I. 15 II. 25 III. 17
Approach: Compute AC for each option and see if it satisfies the basic property relating to
sides of a triangle
Sum of two smaller sides should be greater than the longer side.
5 + 7 + AC = 15
Or AC = 15 – 12 = 3
Is 5 + 3 > 7? Yes.
5 + 7 + AC = 25
Or AC = 25 – 12 = 13
Is 5 + 7 > 13?
I: Perimeter 15 II: Perimeter 25
Which of the following could be the perimeter of the triangle?Side AB measures 5 units and side BC measures 7 units.
Options given: I. 15 II. 25 III. 17
Approach: Compute AC for each option and see if it satisfies the basic property relating to
sides of a triangle
Sum of two smaller sides should be greater than the longer side.
5 + 7 + AC = 15
Or AC = 15 – 12 = 3
Is 5 + 3 > 7? Yes.
5 + 7 + AC = 25
Or AC = 25 – 12 = 13
Is 5 + 7 > 13? No.
I: Perimeter 15 II: Perimeter 25 III: Perimeter 17
Which of the following could be the perimeter of the triangle?Side AB measures 5 units and side BC measures 7 units.
Options given: I. 15 II. 25 III. 17
Approach: Compute AC for each option and see if it satisfies the basic property relating to
sides of a triangle
Sum of two smaller sides should be greater than the longer side.
5 + 7 + AC = 15
Or AC = 15 – 12 = 3
Is 5 + 3 > 7? Yes.
5 + 7 + AC = 25
Or AC = 25 – 12 = 13
Is 5 + 7 > 13? No.
I: Perimeter 15 II: Perimeter 25 III: Perimeter 17
Which of the following could be the perimeter of the triangle?Side AB measures 5 units and side BC measures 7 units.
Options given: I. 15 II. 25 III. 17
Approach: Compute AC for each option and see if it satisfies the basic property relating to
sides of a triangle
Sum of two smaller sides should be greater than the longer side.
5 + 7 + AC = 15
Or AC = 15 – 12 = 3
Is 5 + 3 > 7? Yes.
5 + 7 + AC = 25
Or AC = 25 – 12 = 13
Is 5 + 7 > 13? No.
5 + 7 + AC = 17
I: Perimeter 15 II: Perimeter 25 III: Perimeter 17
Which of the following could be the perimeter of the triangle?Side AB measures 5 units and side BC measures 7 units.
Options given: I. 15 II. 25 III. 17
Approach: Compute AC for each option and see if it satisfies the basic property relating to
sides of a triangle
Sum of two smaller sides should be greater than the longer side.
5 + 7 + AC = 15
Or AC = 15 – 12 = 3
Is 5 + 3 > 7? Yes.
5 + 7 + AC = 25
Or AC = 25 – 12 = 13
Is 5 + 7 > 13? No.
5 + 7 + AC = 17
Or AC = 17 – 12 = 5
I: Perimeter 15 II: Perimeter 25 III: Perimeter 17
Which of the following could be the perimeter of the triangle?Side AB measures 5 units and side BC measures 7 units.
Options given: I. 15 II. 25 III. 17
Approach: Compute AC for each option and see if it satisfies the basic property relating to
sides of a triangle
Sum of two smaller sides should be greater than the longer side.
5 + 7 + AC = 15
Or AC = 15 – 12 = 3
Is 5 + 3 > 7? Yes.
5 + 7 + AC = 25
Or AC = 25 – 12 = 13
Is 5 + 7 > 13? No.
5 + 7 + AC = 17
Or AC = 17 – 12 = 5
Is 5 + 5 > 7?
I: Perimeter 15 II: Perimeter 25 III: Perimeter 17
Which of the following could be the perimeter of the triangle?Side AB measures 5 units and side BC measures 7 units.
Options given: I. 15 II. 25 III. 17
Approach: Compute AC for each option and see if it satisfies the basic property relating to
sides of a triangle
Sum of two smaller sides should be greater than the longer side.
5 + 7 + AC = 15
Or AC = 15 – 12 = 3
Is 5 + 3 > 7? Yes.
5 + 7 + AC = 25
Or AC = 25 – 12 = 13
Is 5 + 7 > 13? No.
5 + 7 + AC = 17
Or AC = 17 – 12 = 5
Is 5 + 5 > 7? Yes.
I: Perimeter 15 II: Perimeter 25 III: Perimeter 17
Correct Answer choice D.Options I and III
Which of the following could be the perimeter of the triangle?Side AB measures 5 units and side BC measures 7 units.
Options given: I. 15 II. 25 III. 17
Approach: Compute AC for each option and see if it satisfies the basic property relating to
sides of a triangle
Sum of two smaller sides should be greater than the longer side.
5 + 7 + AC = 15
Or AC = 15 – 12 = 3
Is 5 + 3 > 7? Yes.
5 + 7 + AC = 25
Or AC = 25 – 12 = 13
Is 5 + 7 > 13? No.
5 + 7 + AC = 17
Or AC = 17 – 12 = 5
Is 5 + 5 > 7? Yes.
Points to Remember
Triangle Properties
Relating to sides of a triangle
Triangle Properties
Relating to sides of a triangle
Sum of any two sides should be greater than the third side
Triangle Properties
For practical reasons one can rewrite the property as
Sum of two smaller sides > longer side
Relating to sides of a triangle
Sum of any two sides should be greater than the third side
For more questions
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