4. The Tangent Ratio
Sign In
Evaluate the missing sides using the tangent ratio and the Pythagorean Theorem.
≈ 128 units
We are asked to evaluate the perimeter of the drawn figure. Let's take a look at the given diagram.
First we are given that AC= 26 and D is the midpoint of AC. This means that DC=AD= 262= 13. Now let's focus on triangle AED.
Notice that we can find ED using the definition of the tangent.Recall that in a right triangle the tangent of an angle is the ratio between the leg opposite to the leg adjacent to this angle. Using this information, let's write an equation for tan50^(∘).
The length of ED is approximately 10.9 units. Now we can find AE using the Pythagorean Theorem. Recall that this theorem tells use that the sum of the squared legs in a right triangle is equal to its squared hypotenuse.
Next let's focus on â–³ ABC. First we will evaluate AB using the tangent ratio.
LHS * 26=RHS* 26
Rearrange equation
Use a calculator
Round to 1 decimal place(s)
The length of AB is approximately 18.2. Now, using the Pythagorean Theorem we will find the length of the hypotenuse BC.
Again notice that since BC represents the side length, we considered only the positive case when taking a square root of BC^2. Let's add the information we found to our diagram.
We know that ∠CBA≅∠FBG≅ ∠HBF. To find the measure of these angles we can use the fact that in a right triangle acute angles are supplementary, so their sum is 90^(∘). Therefore ∠CBA has a measure of 90^(∘)-35^(∘)=55^(∘).
Since we are given that AD=BF, the value of BF is also 13. Notice that â–³ BFG and â–³ BFH are congruent by Angle-Side-Angle Congruence Theorem.
To find GF and FH we will use the tangent ratio for the last time.
LHS * 13=RHS* 13
Rearrange equation
Use a calculator
Round to 1 decimal place(s)
The length of GF and FH is approximately 18.6. Next, using the Pythagorean Theorem we will find the lengths of the hypotenuses of these triangles.
For the last time notice that since BG represents the side length, we considered only the positive case when taking a square root of BG^2. Let's add the information we found to our diagram.
Finally we will find the length of CG. To do this we will evaluate the difference between BC and BG. CG=BC-BG= 31.7- 22.7=9 Now we have all side lengths of the figure ABHFGCDE.
Let's evaluate the perimeter of the figure by adding all side lengths. 17+18.2+22.7+18.6+18.6+9+13+10.9=128 The perimeter of this figure is approximately 128 units. Notice that this value is only an approximation, as we used approximate values to evaluate it.