GMAT Math : Triangles

Study concepts, example questions & explanations for GMAT Math

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Example Questions

Example Question #3 : Calculating The Area Of An Acute / Obtuse Triangle

Which of the following is the area of a triangle on the coordinate plane with its vertices on the points \(\displaystyle (-7,0), (5,0), (M,N)\) , where \(\displaystyle M,N >0\) ?

Possible Answers:

\(\displaystyle 12N\)

\(\displaystyle 6M\)

\(\displaystyle 6N\)

\(\displaystyle 12MN\)

\(\displaystyle 12M\)

Correct answer:

\(\displaystyle 6N\)

Explanation:

We can view the horizontal segment connecting \(\displaystyle (-7,0), (5,0)\), and \(\displaystyle (5,0)\) as the base; its length wiill be \(\displaystyle 5 - (-7) = 12\). The height will be the perpendicular (vertical) distance to this segment from the opposite point \(\displaystyle (M,N)\), which is \(\displaystyle N\), the \(\displaystyle y\)-coordinate; therefore, the area of the triangle will be half the product of these two numbers, or

\(\displaystyle A = \frac{1}{2} \cdot 12 \cdot N = 6N\).

Example Question #111 : Triangles

Give the area of a triangle on the coordinate plane with vertices \(\displaystyle (0,0), (2,6), (5,2)\).

Possible Answers:

\(\displaystyle 12\)

\(\displaystyle 13\)

\(\displaystyle 15\)

\(\displaystyle 10\)

\(\displaystyle 17\)

Correct answer:

\(\displaystyle 13\)

Explanation:

This can be illustrated by showing this triangle inscribed inside a rectangle whose vertices are \(\displaystyle (0,0), (5,0),(0,6),(5.6)\):

Triangle

The area of the white triangle \(\displaystyle A _{W}\) is the one whose area we calculate. To do this, we need the area of the square:

\(\displaystyle A_{\square } = 5 \cdot6 = 30\)

The area of the red triangle:

\(\displaystyle A_{1} = \frac{1}{2} \cdot 5 \cdot 2 = 5\)

The area of the green triangle:

\(\displaystyle A_{2} = \frac{1}{2} \cdot 6 \cdot 2 = 6\)

And the area of the beige triangle:

\(\displaystyle A_{3} = \frac{1}{2} \cdot 4 \cdot 3 = 6\)

The area of the white triangle will be as follows:

\(\displaystyle A _{W} = A_{\square } - \left (A_{1} + A_{2} + A_{3} \right )\)

\(\displaystyle = 30 - (5+6+6)\)

\(\displaystyle = 30 - 17 = 13\)

Example Question #1 : Calculating The Length Of The Side Of An Acute / Obtuse Triangle

Two sides of a triangle measure 5 inches and 11 inches. Which of the following statements correctly expresses the range of possible lengths of the third side \(\displaystyle C\)?

Possible Answers:

\(\displaystyle 6 < C < 16\)

\(\displaystyle 0 < C < 16\)

\(\displaystyle 0 < C < 11\)

\(\displaystyle 5 < C < 11\)

\(\displaystyle C > 6\)

Correct answer:

\(\displaystyle 6 < C < 16\)

Explanation:

By the Triangle Inequality, the sum of the lengths of two shortest sides must exceed that of the third.

Case 1: \(\displaystyle C\) is the greatest of the three sidelengths. 

Then \(\displaystyle C < 5 + 11 = 16\)

Case 2: \(\displaystyle C\) is not the greatest of the three sidelengths - that is, 11 is.

Then \(\displaystyle 11 < C + 5\), or, equivalently, \(\displaystyle 6 < C\).

Therefore, \(\displaystyle 6 < C < 16\).

 

Example Question #1 : Calculating The Length Of The Side Of An Acute / Obtuse Triangle

The sides of a triangle are 4, 8, and an integer \(\displaystyle z\). How many possible values does \(\displaystyle z\) have? 

Possible Answers:

\(\displaystyle 8\)

\(\displaystyle 7\)

\(\displaystyle 6\)

\(\displaystyle 5\)

Correct answer:

\(\displaystyle 7\)

Explanation:

If two sides are 4 and 8, then the third side must be greater than  \(\displaystyle 8-4\) and less than \(\displaystyle 8+4\). This means \(\displaystyle z\) can be 5, 6, 7, 8, 9, 10, or 11. 

Example Question #1 : Calculating The Length Of The Side Of An Acute / Obtuse Triangle

Which of the following is true of a triangle with sides that measure 15, 17, and 21?

Possible Answers:

None of these statements can be proved without further information.

It is a right triangle.

It is an obtuse triangle.

It cannot exist.

It is an acute triangle.

Correct answer:

It is an acute triangle.

Explanation:

The triangle can exist by the Triangle Inequality, since the sum of the two smaller sides exceeds the greatest:

\(\displaystyle 15 + 17 = 32 > 21\)

To determine whether the triangle is acute, right, or obtuse, add the squares of the two smaller sides, and compare the sum to the square of the largest side.

\(\displaystyle 15^{2} + 17^{2} = 225 +289 =514 > 441 = 21^{2}\)

Since this sum is greater, the triangle is acute.

Example Question #4 : Calculating The Length Of The Side Of An Acute / Obtuse Triangle

Let the three interior angles of a triangle measure \(\displaystyle x, x + 40\), and \(\displaystyle x + 80\). Which of the following statements is true about the triangle?

Possible Answers:

The triangle is scalene and acute.

The triangle is scalene and right.

The triangle is scalene and obtuse.

The triangle is isosceles and obtuse.

The triangle is isosceles and acute.

Correct answer:

The triangle is scalene and obtuse.

Explanation:

If these are the measures of the interior angles of a triangle, then they total \(\displaystyle 180^{\circ }\). Add the expressions, and solve for \(\displaystyle x\).

\(\displaystyle x + (x+40) + (x+80) = 180\)

\(\displaystyle 3x + 120 = 180\)

\(\displaystyle 3x = 60\)

\(\displaystyle x=20\)

One angle measures \(\displaystyle x = 20^{\circ }\) The others measure:

\(\displaystyle x+40 = 20 + 40 = 60^{\circ }\)

\(\displaystyle x+80 = 20 + 80 = 100^{\circ }\)

Since the largest angle measures greater than \(\displaystyle 90^{\circ }\), the angle is obtuse, and the triangle is as well. Since the three angles each have different measure, their opposite sides do also, making the triangle scalene.

Example Question #5 : Calculating The Length Of The Side Of An Acute / Obtuse Triangle

In \(\displaystyle \Delta ABC\)\(\displaystyle AB = 7\) and \(\displaystyle BC = 15\). Which of the following values of \(\displaystyle AC\) makes \(\displaystyle \Delta ABC\) a scalene triangle?

Possible Answers:

\(\displaystyle 15\)

\(\displaystyle 8\)

\(\displaystyle 22\)

None of the other responses gives a correct answer.

\(\displaystyle 10\)

Correct answer:

\(\displaystyle 10\)

Explanation:

The three sides of a scalene triangle have different measures, so 15 can be eliminated.

By the Triangle Inequality, the sum of the lengths of the two smaller sides must exceed the length of the third side. Since \(\displaystyle 7+8 =15\), 8 violates this theorem; since \(\displaystyle 7+15 = 22\), 22 does as well. 

10 is a valid measure of the third side, since \(\displaystyle 7 + 10 < 15\); it makes all three segments of different length, so it is the correct choice.

Example Question #2 : Calculating The Length Of The Side Of An Acute / Obtuse Triangle

\(\displaystyle \Delta ABC\) is a scalene triangle with perimeter 33; the length of each of its sides can be given by a prime whole number. What is the greatest possible length of its longest side?

Possible Answers:

\(\displaystyle 19\)

This triangle cannot exist.

\(\displaystyle 17\)

\(\displaystyle 23\)

\(\displaystyle 13\)

Correct answer:

This triangle cannot exist.

Explanation:

By trial and error, we get four ways to add distinct primes to yield sum 33:

\(\displaystyle 3 + 7 + 23\) 

\(\displaystyle 3+ 11 + 19\)

\(\displaystyle 3+ 13+ 17\)

\(\displaystyle 5 + 11 + 17\)

In each case, however the Triangle Inequality is violated - the sum of the two shortest lengths does not exceed the third. 

No triangle can exist as described.

Example Question #2 : Calculating The Length Of The Side Of An Acute / Obtuse Triangle

The lengths of the sides of a scalene triangle are all prime numbers, and so is the perimeter of the triangle. What is the least possible perimeter of the triangle?

Possible Answers:

\(\displaystyle 17\)

\(\displaystyle 23\)

\(\displaystyle 19\)

\(\displaystyle 29\)

\(\displaystyle 13\)

Correct answer:

\(\displaystyle 19\)

Explanation:

A scalene triangle has three sides of different lengths, so we are looking for three distinct prime integers whose sum is a prime integer. 

One of the sides cannot be 2, since the sum of 2 and two odd primes would be an even number greater than 2, a composite number. Therefore, beginning with the least three odd primes, add increasing triples of distinct prime numbers, as follows, until a solution presents itself:

\(\displaystyle 3+5+7 =15\) - incorrect

\(\displaystyle 3+5+11 =19\) - correct

The correct answer, 19, presents itself quickly.

Example Question #3 : Calculating The Length Of The Side Of An Acute / Obtuse Triangle

\(\displaystyle \Delta ABC\) is a scalene triangle with perimeter 47; the length of each of its sides can be given by a prime whole number. What is the greatest possible length of its longest side?

Possible Answers:

This triangle cannot exist.

\(\displaystyle 17\)

\(\displaystyle 27\)

\(\displaystyle 19\)

\(\displaystyle 23\)

Correct answer:

\(\displaystyle 23\)

Explanation:

A scalene triangle has three sides of different lengths, so we are looking for three distinct prime integers whose sum is 47. 

There are ten ways to add three distinct primes to yield sum 47:

\(\displaystyle 3 + 5+39\)

\(\displaystyle 3+7+37\)

\(\displaystyle 3+13+31\)

\(\displaystyle 5+11+31\)

\(\displaystyle 5+13+29\)

\(\displaystyle 5+19+23\)

\(\displaystyle 7 + 11+29\)

\(\displaystyle 7 + 17+23\)

\(\displaystyle 11+13+23\)

\(\displaystyle 11+17+19\)

By the Triangle Inequality, the sum of the lengths of the shortest two sides must exceed that of the greatest. We can therefore eliminate all but four:

\(\displaystyle 5+19+23\)

\(\displaystyle 7 + 17+23\)

\(\displaystyle 11+13+23\)

\(\displaystyle 11+17+19\)

The greatest possible length of the longest side is 23.

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