ACT Math : How to find the length of the side of of an acute / obtuse isosceles triangle

Study concepts, example questions & explanations for ACT Math

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Example Questions

Example Question #11 : Acute / Obtuse 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:


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 #231 : Act Math

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:


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 #232 : Act Math


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:


Because  is isosceles,  equals  or .

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

Example Question #233 : Act Math

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:


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.

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