ACT Math : Triangles

Study concepts, example questions & explanations for ACT Math

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Example Questions

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Example Question #2 : How To Find An Angle In An Acute / Obtuse Isosceles Triangle

The base angle of an isosceles triangle is thirteen more than three times the vertex angle. What is the difference between the vertex angle and the base angle?

Possible Answers:

Correct answer:

Explanation:

Every triangle has . An isosceles triangle has one vertex ange, and two congruent base angles.

Let  be the vertex angle and  be the base angle.

The equation to solve becomes , since the base angle occurs twice.

Now we can solve for the vertex angle.

The difference between the vertex angle and the base angle is .

Example Question #3 : How To Find An Angle In An Acute / Obtuse Isosceles Triangle

A particular acute isosceles triangle has an internal angle measuring . Which of the following must be the other two angles?

Possible Answers:

Correct answer:

Explanation:

By definition, an acute isosceles triangle will have at least two sides (and at least two corresponding angles) that are congruent, and no angle will be greater than . Addtionally, like all triangles, the three angles will sum to . Thus, of our two answers which sum to , only  is valid, as  would violate the "acute" part of the definition.

Example Question #4 : How To Find An Angle In An Acute / Obtuse Isosceles Triangle

In triangle ABC, Angle A = x degrees, Angle B = 2x degrees, and Angle C = 3x+30 degrees. How many degrees is Angle B?

Possible Answers:

50°

105°

45°

25°

30°

Correct answer:

50°

Explanation:

Because the interior angles of a triangle add up to 180°, we can create an equation using the variables given in the problem: x+2x+(3x+30)=180. This simplifies to 6X+30=180. When we subtract 30 from both sides, we get 6x=150. Then, when we divide both sides by 6, we get x=25. Because Angle B=2x degrees, we multiply 25 times 2. Thus, Angle B is equal to 50°. If you got an answer of 25, you may have forgotten to multiply by 2. If you got 105, you may have found Angle C instead of Angle B.

Example Question #11 : Isosceles Triangles

A triangle has a perimeter of  inches with one side of length  inches. If the remaining two sides have lengths in a ratio of , what is length of the shortest side of the triangle?

Possible Answers:

Correct answer:

Explanation:

The answer is .

Since we know that the permieter is  inches and one side is  inches, it can be determined that the remaining two sides must combine to be  inches. The ratio of the remaining two sides is  which means 3 parts : 4 parts or 7 parts combined. We can then set up the equation , and divide both sides by  which means . The ratio of the remaining side lengths then becomes  or . We now know the 3 side lengths are .

 is the shortest side and thus the answer.

Example Question #1 : How To Find The Length Of The Side Of Of An Acute / Obtuse Isosceles Triangle

In the standard  coordinate plane, the points  and  form two vertices of an isosceles triangle.  Which of the following points could be the third vertex? 

Possible Answers:

Correct answer:

Explanation:

To form an isosceles triangle here, we need to create a third vertex whose  coordinate is between  and .  If a vertex is placed at , the distance from  to this point will be . The distance from  to this point will be the same.

Example Question #2 : How To Find The Length Of The Side Of Of An Acute / Obtuse Isosceles Triangle

Screen_shot_2015-05-07_at_4.37.54_pm

Note: Figure is not drawn to scale.

In the figure above, points  are collinear and   is a right angle. If  and  is , what is ?  

Possible Answers:

Correct answer:

Explanation:

Because  is isosceles,  equals  or .

We know that  add up to , so  must equal  or .

Example Question #3 : How To Find The Length Of The Side Of Of An Acute / Obtuse Isosceles Triangle

A light beam of pure white light is aimed horizontally at a prism, which splits the light into two streams that diverge at a  angle. The split beams each travel exactly  from the prism before striking two optic sensors (one for each beam).

What is the distance, in feet, between the two sensors?

Round your final answer to the nearest tenth. Do not round until then.

Possible Answers:

Correct answer:

Explanation:

This problem can be solved when one realizes that the light beam's split has resulted in an acute isosceles triangle. The triangle as stated has two sides of  feet apiece, which meets the requirement for isosceles triangles, and having one angle of  at the vertex where the two congruent sides meet means the other two angles must be  and . The missing side connecting the two sensors, therefore, is opposite the  angle.

Since we know at least two angles and at least one side of our triangle, we can use the Law of Sines to calculate the remainder. The Law of Sines says that for any triangle with angles  and  and opposite sides  and :

.

Plugging in one of our  angles (and its corresponding  ft side) into this equation, as well as our  angle (and its corresponding unknown side) into this equation gives us:

Next, cross-multiply:

 ---> 

Now simplify and solve:

Rounding, we see our missing side is  long.

Example Question #1 : Acute / Obtuse Isosceles Triangles

An isosceles triangle has a base of 12\ cm and an area of 42\ cm^{2}. What must be the height of this triangle?

Possible Answers:

10\ cm

7\ cm

9\ cm

8\ cm

6\ cm

Correct answer:

7\ cm

Explanation:

A=\frac{1}{2}bh

6x=42

x=7

Example Question #1 : How To Find The Height Of Of An Acute / Obtuse Isosceles Triangle

What is the height of an isosceles triangle which has a base of  and an area of ?

Possible Answers:

Correct answer:

Explanation:

The area of a triangle is given by the equation: 

In this case, we are given the area () and the base () and are asked to solve for height ().

To do this, we must plug in the given values for  and which gives the following:

We then must multiply the right side, and then divide the entire equation by 2, in order to solve for :

Therefore, the height of the triangle is .

Example Question #1 : How To Find If Of Acute / Obtuse Isosceles Triangles Are Congruent

There are two obtuse triangles. The obtuse angle of triangle one is . The sum of two angles in the second triangle is . When are these two triangles congruent?

Possible Answers:

The two triangles must be congruent

When the sum of angle A and angle B in triangle 1 is equal to the sum of the corresponding angles in triangle 2

When the obtuse angle is congruent to the smallest angle of the other triangle

Cannot be determined

The two triangles cannot be congruent

Correct answer:

The two triangles cannot be congruent

Explanation:

In order for two obtuse triangles to be congruent, the sum of the two smaller angles must equal the sum of the two smaller angles of the second triangle. That is, excluding the obtuse angle. 

The first triangle has an obtuse angle of . That means the sum of the other two angles is . The sum of the corresponding angles in triangle 2 is . Therefore, because  is not equal to , the two obtuse triangles cannot be congruent.

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