PSAT Math : Triangles

Study concepts, example questions & explanations for PSAT Math

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Example Questions

Example Question #1 : How To Find The Length Of The Hypotenuse Of An Acute / Obtuse 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?

 

Possible Answers:

4

4√3

3√3

5

6

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 #62 : 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:
2
12
11
3
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 #82 : Triangles

A triangle with two equal angles is called a(n) __________.

Possible Answers:

equilateral triangle

disjoint triangle

Pythagoras triangle

isosceles triangle

right triangle

Correct answer:

isosceles triangle

Explanation:

An isoceles triangle is a triangle that has at least two congruent sides (and therefore, at least two congruent angles as well).

Example Question #1 : How To Find The Perimeter Of A 45/45/90 Right Isosceles Triangle

An isosceles triangle has a base of 6 and a height of 4. What is the perimeter of the triangle?

Possible Answers:

None of these

Correct answer:

Explanation:

An isosceles triangle is basically two right triangles stuck together. The isosceles triangle has a base of 6, which means that from the midpoint of the base to one of the angles, the length is 3. Now, you have a right triangle with a base of 3 and a height of 4. The hypotenuse of this right triangle, which is one of the two congruent sides of the isosceles triangle, is 5 units long (according to the Pythagorean Theorem).

The total perimeter will be the length of the base (6) plus the length of the hypotenuse of each right triangle (5).

5 + 5 + 6 = 16

Example Question #1 : 45/45/90 Right Isosceles Triangles

What is the area of a square that has a diagonal whose endpoints in the coordinate plane are located at (-8, 6) and (2, -4)?

Possible Answers:

100

50√2

100√2

50

200√2

Correct answer:

100

Explanation:

Square_part1

Square_part2

Square_part3

Example Question #81 : Right Triangles

Three points in the xy-coordinate system form a triangle.

The points are .

What is the perimeter of the triangle?

Possible Answers:

9 + \sqrt{26}

9 + \sqrt{41}

9 + \sqrt{71}

Correct answer:

9 + \sqrt{41}

Explanation:

Drawing points gives sides of a right triangle of 4, 5, and an unknown hypotenuse.

Using the pythagorean theorem we find that the hypotenuse is \sqrt{41}.

Example Question #82 : Right Triangles

Triangle

Based on the information given above, what is the perimeter of triangle ABC?

Possible Answers:

Correct answer:

Explanation:

Triangle-solution

Consult the diagram above while reading the solution. Because of what we know about supplementary angles, we can fill in the inner values of the triangle. Angles A and B can be found by the following reductions:

A + 120 = 180; A = 60

B + 150 = 180; B = 30

Since we know A + B + C = 180 and have the values of A and B, we know:

60 + 30 + C = 180; C = 90

This gives us a 30:60:90 triangle. Now, since 17.5 is across from the 30° angle, we know that the other two sides will have to be √3 and 2 times 17.5; therefore, our perimeter will be as follows:

Example Question #1 : Right Triangles

Triangles

In the figure above, line segments DC and AB are parallel. What is the perimeter of quadrilateral ABCD?

Possible Answers:

75

85

90

80

95

Correct answer:

85

Explanation:

Because DC and AB are parallel, this means that angles CDB and ABD are equal. When two parallel lines are cut by a transversal line, alternate interior angles (such as CDB and ABD) are congruent.

Now, we can show that triangles ABD and BDC are similar. Both ABD and BDC are right triangles. This means that they have one angle that is the same—their right angle. Also, we just established that angles CDB and ABD are congruent. By the angle-angle similarity theorem, if two triangles have two angles that are congruent, they are similar. Thus triangles ABD and BDC are similar triangles.

We can use the similarity between triangles ABD and BDC to find the lengths of BC and CD. The length of BC is proportional to the length of AD, and the length of CD is proportional to the length of DB, because these sides correspond.

We don’t know the length of DB, but we can find it using the Pythagorean Theorem. Let a, b, and c represent the lengths of AD, AB, and BD respectively. According to the Pythagorean Theorem:

 a2 + b2 = c2

152 + 202 = c2

625 = c2

c = 25

The length of BD is 25.

Similar_triangles

We now have what we need to find the perimeter of the quadrilateral.

Perimeter = sum of the lengths of AB, BC, CD, and DA.

Perimeter = 20 + 18.75 + 31.25 + 15 = 85

The answer is 85. 

Example Question #11 : Triangles

 and  is a right angle.

Which angle or angles must be complementary to  ?

I) 

II) 

III) 

IV) 

V) 

Possible Answers:

II only

I and III only

II and V only

I only

IV only

Correct answer:

II and V only

Explanation:

 is a right angle, and, since corresponding angles of similar triangles are congruent, so is . A right angle cannot be part of a complementary pair so both can be eliminated.

 can be eliminated, since it is congruent to ; congruent angles are not necessarily complementary.

Since  is right angle,  is a right triangle, and  and  are its acute angles. That makes  complementary to . Since  is congruent to , it is also complementary to .

The correct response is II and V only.

Example Question #12 : Triangles

Triangles

Refer to the above figure. Given that , give the perimeter of .

Possible Answers:

Correct answer:

Explanation:

By the Pythagorean Theorem, 

 

The similarity ratio of  to  is 

which is subsequently the ratio of the perimeter of   to that of .

The perimeter of  is 

,

so the perimeter of  can be found using this ratio:

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