SAT Math : Isosceles Triangles

Study concepts, example questions & explanations for SAT Math

varsity tutors app store varsity tutors android store varsity tutors ibooks store

Example Questions

Example Question #1 : Isosceles Triangles

In an isosceles triangle, the vertex angle is 15 less than the base angle.  What is the base angle?

Possible Answers:

Correct answer:

Explanation:

Every triangle has 180 degrees.  An isosceles triangle has one vertex angle and two congruent base angles.

Let  = base angle and  = vertex angle

So the equation to solve becomes

Thus, 65 is the base angle and 50 is the vertex angle.

Example Question #2 : Isosceles Triangles

In an isosceles triangle the vertex angle is half the base angle.  What is the vertex angle?

Possible Answers:

108

72

54

36

45

Correct answer:

36

Explanation:

Every triangle has 180 degrees.  An isosceles triangle has one vertex angle and two congruent base angles.

Let x = base angle and 0.5x = vertex angle

So the equation to solve becomes x+x+0.5x=180, thus x=72 is the base angle and 0.5x=36 is the vertex angle.

Example Question #3 : Isosceles Triangles

If the average (arithmetic mean) of two noncongruent angles of an isosceles triangle is , which of the following is the measure of one of the angles of the triangle?

Possible Answers:

Correct answer:

Explanation:

Since the triangle is isosceles, we know that 2 of the angles (that sum up to 180) must be equal. The question states that the noncongruent angles average 55°, thus providing us with a system of two equations:

Solving for x and y by substitution, we get x = 70° and y = 40° (which average out to 55°).

70 + 70 + 40 equals 180 also checks out.

Since 70° is not an answer choice for us, we know that the 40° must be one of the angles.

Example Question #4 : Isosceles Triangles

The base angle of an isosceles triangle is 27^{\circ}.  What is the vertex angle?

Possible Answers:

108^{\circ}

75^{\circ}

126^{\circ}

149^{\circ}

135^{\circ}

Correct answer:

126^{\circ}

Explanation:

Every triangle has 180 degrees.  An isosceles triangle has one vertex angle and two congruent base angles. 

Solve the equation 27+27+x=180 for x to find the measure of the vertex angle. 

x = 180 - 27 - 27

x = 126

Therefore the measure of the vertex angle is 126^{\circ}.

Example Question #11 : Isosceles Triangles

An isosceles triangle has an area of 12. If the ratio of the base to the height is 3:2, what is the length of the two equal sides?

 

Possible Answers:

3√3

6

4

5

4√3

Correct answer:

5

Explanation:

Area of a triangle is ½ x base x height. Since base:height = 3:2, base = 1.5 height.  Area = 12 = ½ x 1.5 height x height or 24/1.5 = height2.  Height = 4.  Base = 1.5 height = 6. Half the base and the height form the legs of a right triangle, with an equal leg of the isosceles triangle as the hypotenuse. This is a 3-4-5 right triangle.

 Sat_math_167_01

 

 

 

 

Example Question #12 : Isosceles Triangles

Two sides of a triangle each have length 6. All of the following could be the length of the third side EXCEPT

Possible Answers:
11
2
3
12
1
Correct answer: 12
Explanation:

This question is about the Triangle Inequality, which states that in a triangle with two sides A and B, the third side must be greater than the absolute value of the difference between A and B and smaller than the sum of A and B.

Applying the Triangle Inequality to this problem, we see that the third side must be greater than the absolute value of the difference between the other two sides, which is |6-6|=0, and smaller than the sum of the two other sides, which is 6+6=12. The only answer choice that does not satisfy this range of possible values is 12 since the third side must be LESS than 12.

 

Example Question #13 : Isosceles Triangles

A triangle has the following side lengths:

Which of the following correctly describes the triangle? 

Possible Answers:

Obtuse and isosceles

Acute and isosceles

None of these

Obtuse and scalene

Acute and scalene

Correct answer:

Acute and isosceles

Explanation:

The triangle has two sides of equal length, 13, so it is by definition isosceles. 

To determine whether the triangle is acute, right, or obtuse, compare the sum of the squares of the lengths of the two shortest sides to the square of the length of the longest side. The former quantity is equal to

The latter quantity is equal to

The former is greater than the latter; consequently, the triangle is acute. The correct response is that the triangle is acute and isosceles.

Learning Tools by Varsity Tutors