How to find the length of the side of an acute / obtuse triangle - Intermediate Geometry

This problem requires us to use either the Law of Sines or the Law of Cosines. To figure out which one we should use, let's write down all the information we have in this format:

A = 35° a = ?

B = ? b = 8

C = ? c = 12

Now we can easily see that we do not have any complete pairs (such as A and a, or B and b) but we do have one term from each pair (A, b, and c). This tells us that we should use the Law of Cosines. (We use the Law of Sines when we have one complete pair, such as B and b).

Law of Cosines:

Since we are solving for a, let's use the first version of the Law of Cosines.
This gives us:

This problem requires us to use either the Law of Sines or the Law of Cosines. To figure out which one we should use, let's write down all the information we have in this format:

A = 25° a = 17

B = 50° b = ?

C = ? c = ?

Now we can easily see that we have a complete pair, A and a. This tells us that we can use the Law of Sines. (We use the Law of Cosines when we do not have a complete pair).

Law of Sines:

To solve for b, we can use the first two terms which gives us:

Use the pythagorean theorem:

The triangle has two sides of the same length, 18.4, so, by definition, it is isosceles.

Convert each of the three measures to the same unit; we will choose the smallest unit, millimeters.

One meter is equivalent to 1,000 millimeters, so 12 meters can be converted to millimeters by multiplying by 1,000:

One centimeter is equivalent to ten millimeters, so 1,200 cenitmeters can be converted to millimeters by multiplying by 10:

These two sides have the same length. However, the third side, which has length 12 millimeters, is of different length. Since the triangle has exactly two congruent sides, it is by definition isosceles, but not equilateral.

Convert each of the three measures to the same unit; we will choose the smallest unit, inches.

One foot is equal to twelve inches, so feet can be converted to inches by multiplying by 12:

One yard is equal to 36 inches.

The lengths of the sides in inches are 18, 24, and 36. Since no two sides have the same measure, the triangle is by definition scalene.

The measures of the interior angles of a triangle add up to . If is the measure of the third angle, then

Solve for :

By the Isosceles Triangle Theorem, if two sides of a triangle have the same length, their opposite angles have the same measure. Since no two angles have the same measure, no two sides have the same length. This makes the triangle scalene.

The measures of the interior angles of a triangle add up to . If is the measure of the third angle, then

Solve for :

The triangle has two congruent angles - each with measure . As a consequence, by the Converse of the Isosceles Triangle Theorem, the triangle has two congruent sides, making it, by definition, isosceles.

The measures of the interior angles of a triangle add up to . If is the measure of the third angle, then

Solve for :

The triangle therefore has an angle with measure greater than - an obtuse angle. The triangle is by definition an obtuse triangle,

At a given vertex, an exterior angle and an interior angle of a triangle form a linear pair, making them supplementary - that is, their measures total . The measures of two interior angles can be calculated by subtracting the exterior angle measures from :

The triangle has two interior angles of measure .

By the Converse of the Isosceles Triangle Theorem, since the triangle has two congruent angles, their opposite sides are congruent. Also, since an equilateral triangle has three angles of measure , the triangle is not equilateral. The triangle is isosceles, but not equilateral.

First, convert all of the given sidelengths to the same unit; here, we choose the smallest unit, millimeters.

One meter is equal to 1,000 millimeters.

One centimeter is equal to 10 millimeters, so convert 250 centimeters to millimeters by multiplying by 10:

The measures of the sides of the triangle, in millimeters, are 1,000, 1,200, and 2,500.

By the Triangle Inequality, the sum of the measures of the shortest two sides of a triangle must exceed the length of the longest side, so for this triangle to be possible it must hold that

or

This is false, so a triangle with these sidelengths cannot exist.

First, convert all of the given sidelengths to the same unit; here, we choose the smallest unit, inches.

One foot is equal to 12 inches, so to convert feet to inches, multiply by 12:

One yard is equal to 36 inches, so to convert yards to inches, multiply by 36:

The measures of the sides of the triangle, in inches, are 50, 54, and 60.

By the Triangle Inequality Theorem, the sum of the measures of the shortest two sides of a triangle must exceed the length of the longest side, so for this triangle to be possible it must hold that

or

This is indeed the case, so a triangle with these sidelengths can exist.

By the Triangle Inequality Theorem, the sum of the measures of the shortest two sides of a triangle must exceed the length of the longest side.

Write each length in terms of a common denominator; this is . The fractions convert as follows:

is the greatest of the three, so for this triangle to be possible it must hold that

or, equivalently,

This is indeed the case, so a triangle with these sidelengths can exist.

By the Triangle Inequality Theorem, the sum of the measures of the shortest two sides of a triangle must exceed the length of the longest side.

Write each length in terms of a common denominator; this is . The fractions convert as follows:

is the greatest of the three, so for this triangle to be possible it must hold that

or, equivalently,

This is false, so a triangle with these sidelengths cannot exist.

In addition to the fact that and , we also have that , since, by the Reflexive Property of Congruence, any segment is congruent to itself. We can restate this in a more usable form as .

Therefore, two sides of are congruent to two corresponding sides of , but the included angle of the former has greater measure than that of the latter. It follows from the Hinge Theorem, or Side-Angle-Side Inequality Theorem, that the third side of the former is longer than the third side of the latter - that is, . The statement is true.

The sum of the measures of the interior angles of a triangle is , so

Substitute the given two angle measures and solve for :

Subtract from both sides:

Therefore,

By the Isosceles Triangle Theorem, if , their opposite sides are also congruent - that is, . Since this is not the case, .

We are given that two sides of a triangle, sides and of , are congruent to two sides of another triangle, sides and of ; we are also given that the included angle of the former, , has greater degree measure than that of the latter, . It is a consequence of the Hinge Theorem (also known as the Side-Angle-Side Inequality Theorem) that the side opposite the former is longer than that opposite the latter - that is, .

In any triangle, the sum of the lengths of any two sides is greater than the length of the third side; the statement is a specific example. This is a direct result of the Triangle Inequality Theorem.

The triangle cannot be constructed because the sum of the lengths of the two shorter sides does not exceed the length of the longest side. This violates the conditions of the Triangle Inequality Theorem.