Acute / Obtuse Isosceles Triangles
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PSAT Math › Acute / Obtuse Isosceles Triangles
The base angle of an isosceles triangle is 15 less than three times the vertex angle. What is the vertex angle?
Explanation
Every triangle contains 180 degrees. An isosceles triangle has one vertex angle and two congruent base angles.
Let 
So the equation to solve becomes 
The base angle of an isosceles triangle is 15 less than three times the vertex angle. What is the vertex angle?
Explanation
Every triangle contains 180 degrees. An isosceles triangle has one vertex angle and two congruent base angles.
Let
So the equation to solve becomes
The base angle of an isosceles triangle is five more than twice the vertex angle. What is the base angle?
Explanation
Every triangle has 180 degrees. An isosceles triangle has one vertex angle and two congruent base angles.
Let 
So the equation to solve becomes
Thus the vertex angle is 34 and the base angles are 73.
The base angle of an isosceles triangle is five more than twice the vertex angle. What is the base angle?
Explanation
Every triangle has 180 degrees. An isosceles triangle has one vertex angle and two congruent base angles.
Let
So the equation to solve becomes
Thus the vertex angle is 34 and the base angles are 73.
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?
4
5
3√3
6
4√3
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.
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?
4
5
3√3
6
4√3
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.
Two sides of an isosceles triangle are 20 and 30. What is the difference of the largest and the smallest possible perimeters?
The answer cannot be determined
0
30
15
10
Explanation
The trick here is that we don't know which is the repeated side. Our possible triangles are therefore 20 + 20 + 30 = 70 or 30 + 30 + 20 = 80. The difference is therefore 80 – 70 or 10.
Two sides of an isosceles triangle are 20 and 30. What is the difference of the largest and the smallest possible perimeters?
The answer cannot be determined
0
30
15
10
Explanation
The trick here is that we don't know which is the repeated side. Our possible triangles are therefore 20 + 20 + 30 = 70 or 30 + 30 + 20 = 80. The difference is therefore 80 – 70 or 10.
Two sides of a triangle each have length 6. All of the following could be the length of the third side EXCEPT
1
2
3
11
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.
Two sides of a triangle each have length 6. All of the following could be the length of the third side EXCEPT
1
2
3
11
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.