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ACT Math : Isosceles Triangles

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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 (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:

(4,5)

(2,3)

(2,4)

(0,0)

(3,4)

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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Example Question #1 : 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:

$8\ cm$

$9\ cm$

$6\ cm$

$10\ cm$

$7\ cm$

Correct answer:

$7\ cm$

Explanation:

$A=\frac{1}{2}bh$ .

$6x=42$

$x=7$

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Example Question #231 : Plane Geometry

What is the height of an isosceles triangle which has a base of 4cm and an area of 24cm^2 ?

Possible Answers:

12cm

16cm

6cm

8cm

24cm

Correct answer:

12cm

Explanation:

The area of a triangle is given by the equation:

$\small A=\frac{1}{2}bh$

In this case, we are given the area ( $\small A$ ) and the base ( $\small b$ ) and are asked to solve for height ( $\small h$ ).

To do this, we must plug in the given values for $\small A$ and $\small b$ , which gives the following:

$\small 24\;cm^2=\frac{1}{2}(4\; cm)h$

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

$\small 24\;cm^2=2\;cm(h)$

$\small \frac{24\; cm^2}{2\; cm}=\frac{2\; cm(h)}{2\; cm}$

$\small 12\;cm=h$

$\small h=12\;cm$

Therefore, the height of the triangle is $\small 12\;cm$ .

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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 $101^{\circ}$ . The sum of two angles in the second triangle is $80^{\circ}$ . When are these two triangles congruent?

Possible Answers:

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

The two triangles cannot 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

Cannot be determined

The two triangles must 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 $101^{\circ}$ . That means the sum of the other two angles is $79^{\circ}$ . The sum of the corresponding angles in triangle 2 is $80^{\circ}$ . Therefore, because $79^{\circ}$ is not equal to $80^{\circ}$ , the two obtuse triangles cannot be congruent.

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ACT Math : Isosceles Triangles

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

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

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

Example Question #1 : Triangles

Example Question #231 : Plane Geometry

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

All ACT Math Resources

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