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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