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

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Question

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?

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 $12,12,and\ 16$ .

is the shortest side and thus the answer.

Compare your answer with the correct one above

Question

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?

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 $\sqrt{17}$ . The distance from to this point will be the same.

Compare your answer with the correct one above

Question

Note: Figure is not drawn to scale.

In the figure above, points are collinear and $\angle$ is a right angle. If $\overline{BD}=\overline{DE}$ and $\measuredangle$ is $58^{\circ}$ , what is $\measuredangle CBD$ ?

Answer

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 $29^{\circ}$ .

Compare your answer with the correct one above

Question

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.

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 $30^{\circ}$ 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 and and opposite sides 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.

Compare your answer with the correct one above

Question

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?

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 $12,12,and\ 16$ .

is the shortest side and thus the answer.

Compare your answer with the correct one above

Question

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?

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 $\sqrt{17}$ . The distance from to this point will be the same.

Compare your answer with the correct one above

Question

Note: Figure is not drawn to scale.

In the figure above, points are collinear and $\angle$ is a right angle. If $\overline{BD}=\overline{DE}$ and $\measuredangle$ is $58^{\circ}$ , what is $\measuredangle CBD$ ?

Answer

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 $29^{\circ}$ .

Compare your answer with the correct one above

Question

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.

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 $30^{\circ}$ 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 and and opposite sides 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.

Compare your answer with the correct one above

Question

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?

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 $12,12,and\ 16$ .

is the shortest side and thus the answer.

Compare your answer with the correct one above

Question

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?

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 $\sqrt{17}$ . The distance from to this point will be the same.

Compare your answer with the correct one above

Question

Note: Figure is not drawn to scale.

In the figure above, points are collinear and $\angle$ is a right angle. If $\overline{BD}=\overline{DE}$ and $\measuredangle$ is $58^{\circ}$ , what is $\measuredangle CBD$ ?

Answer

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 $29^{\circ}$ .

Compare your answer with the correct one above

Question

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.

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 $30^{\circ}$ 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 and and opposite sides 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.

Compare your answer with the correct one above

Question

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?

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 $12,12,and\ 16$ .

is the shortest side and thus the answer.

Compare your answer with the correct one above

Question

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?

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 $\sqrt{17}$ . The distance from to this point will be the same.

Compare your answer with the correct one above

Question

Note: Figure is not drawn to scale.

In the figure above, points are collinear and $\angle$ is a right angle. If $\overline{BD}=\overline{DE}$ and $\measuredangle$ is $58^{\circ}$ , what is $\measuredangle CBD$ ?

Answer

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 $29^{\circ}$ .

Compare your answer with the correct one above

Question

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.

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 $30^{\circ}$ 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 and and opposite sides 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.

Compare your answer with the correct one above

Question

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?

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 $12,12,and\ 16$ .

is the shortest side and thus the answer.

Compare your answer with the correct one above

Question

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?

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 $\sqrt{17}$ . The distance from to this point will be the same.

Compare your answer with the correct one above

Question

Note: Figure is not drawn to scale.

In the figure above, points are collinear and $\angle$ is a right angle. If $\overline{BD}=\overline{DE}$ and $\measuredangle$ is $58^{\circ}$ , what is $\measuredangle CBD$ ?

Answer

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 $29^{\circ}$ .

Compare your answer with the correct one above

Question

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.

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 $30^{\circ}$ 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 and and opposite sides 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.

Compare your answer with the correct one above

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