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

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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 3:4 , 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 40-12=28 inches. The ratio of the remaining two sides is 3:4 which means 3 parts : 4 parts or 7 parts combined. We can then set up the equation 7x=28 , and divide both sides by which means x=4 . The ratio of the remaining side lengths then becomes 3*(4):4*(4) or 12:16 . We now know the 3 side lengths are $12,12,and\ 16$ .

is the shortest side and thus the answer.

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Example Question #21 : Acute / Obtuse Isosceles Triangles

In the standard (x,y) coordinate plane, the points (1,-1) and (1,7) form two vertices of an isosceles triangle. Which of the following points could be the third vertex?

Possible Answers:

(2,4)

(3,4)

(4,5)

(2,3)

(0,0)

Correct answer:

(2,3)

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 (2,3) , the distance from (1,-1) to this point will be $\sqrt{17}$ . The distance from (1,7) to this point will be the same.

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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 A, B, C are collinear and $\angle$ ABE is a right angle. If $\overline{BD}=\overline{DE}$ and $\measuredangle$ BDE is $58^{\circ}$ , what is $\measuredangle CBD$ ?

Possible Answers:

$35^{\circ}$

$61^{\circ}$

$29^{\circ}$

$30^{\circ}$

$122^{\circ}$

Correct answer:

$29^{\circ}$

Explanation:

Because $\bigtriangleup BDE$ is isosceles, $\measuredangle DBE$ equals $\left(\frac{180-58}{2}\right)^{\circ}$ or $61^{\circ}$ .

We know that $\measuredangle ABE, \measuredangle DBE, \measuredangle CBD$ add up to $180^{\circ}$ , so $\measuredangle CBD$ must equal $(180-90-61)^{\circ}$ or $29^{\circ}$ .

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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 $30^{\circ}$ angle. The split beams each travel exactly $\textup{5 feet}$ 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:

2.9

2.1

7.3

7.7

2.6

Correct answer:

2.6

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 $30^{\circ}$ at the vertex where the two congruent sides meet means the other two angles must be $75^{\circ}$ and $75^{\circ}$ . The missing side connecting the two sensors, therefore, is opposite the $30^{\circ}$ 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 A, B and and opposite sides a, b and :

$\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}$ .

Plugging in one of our $75^{\circ}$ angles (and its corresponding ft side) into this equation, as well as our $30^{\circ}$ angle (and its corresponding unknown side) into this equation gives us:

$\frac{\sin 75^{\circ}}{5} = \frac{ \sin 30^{\circ}}{x}$

Next, cross-multiply:

$\frac{\sin 75^{\circ}}{5} = \frac{\sin 30^{\circ}}{x}$ ---> $x\sin 75^{\circ} = 5\sin 30^{\circ}$

Now simplify and solve:

$x = \frac{5\sin 30^{\circ}}{\sin 75^{\circ}} = 2.588$

Rounding, we see our missing side is $2.65\textup{ feet}$ long.

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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

Example Question #21 : Acute / Obtuse Isosceles Triangles

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

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

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