# ACT Math : Acute / Obtuse Isosceles Triangles

## Example Questions

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### 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?

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

Note: Figure is not drawn to scale.

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

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.

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 : Isosceles Triangles

An isosceles triangle has a base of  and an area of . What must be the height of this triangle?

Explanation:

### 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 ?

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?

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