2.4  Solving Right Triangles

 

          An exact number is the result of counting or theoretical work, but NOT measured.

 

The result of a calculation can be no more accurate than the least accurate number in the calculation.

          (The chain is only as strong as its weakest link.)

 

When measure a length to “nearest foot” as 7 ft

– the measurement is between 6.5 and 7.5

 

When measure to nearest tenth as 7.4 feet

– the measurement is between 7.35 and 7.45

 

Significant Digits – the digits obtain from measurement

          607 has 3 significant digits            

43.1 has 3 significant digits

          52.00 has 4 significant digits         

.0029 has 2 significant digits (the zeros are place holders)

          .003160 has 4 significant digits

         (the first 2 zeros are placeholders, but the last zero was measured)

          4,300 has 2, 3, or 4 (but defaults to 2) significant digits

                   unless it is known how accurately it was measured.

 

It may help if consider renaming the number as scientific notation 

 .003160 = 3.160 X 10-3

 

Angle Measure to Nearest

Example

Number of Significant Digits

Degree

62o ,  36o

2

Ten minutes, or nearest tenth of a degree

52o 30’’,  60.4o

3

Minute, or nearest hundredth of a degree

81o 48’, 71.25o

4

Ten seconds, or nearest thousandth of a degree

10o 51’ 20”,  21.264o

5

 

Remember – The answer is no more accurate than the least accurate number in the calculations.

 

Solving Right Triangles

Solving a triangle means to find the measure of all three sides and all three angles.

Notation:    side a is opposite angle A;

side b is opposite angle B, and

side c (the hypotenuse) is opposite the right angle C

 

Comment:   It is better to use the original information as much as possible, rather than using intervening values.  However using intermediate values can provide an excellent check.

 

 

Example

          Solve a right triangle if B = 24o 40’,  a = 25.3 cm      

 

 

Again, for greatest accuracy, use the given information as much as possible and avoid rounding off in intermediate steps.

 

Solve right triangle ABC, if a = 44.25 cm  and  b = 44.87 cm

 

 

Angles of Elevation or Depression

 

 

 

 

The angle of elevation is the angle with one side horizontal and the other the ray to the higher point.

 

The angle of depression is the angle with one side horizontal and the other the ray to the lower point.  

 

 

Note:  Congruent angles from alternate interior angles may be helpful.

 

 

Example

The angle of depression from the top of a tree to a point on the ground 15.5 m from the base of the tree is 60.4o.  Find the height of the tree.