Math Problem Statement

Solve the right triangle where angle F = 22°, FG = 8, and angle G is the right angle. Find GH, FG, and m∠H.

Solution

In the image, the problem involves solving a right triangle HGF\triangle HGF with the following information given:

  • F=22\angle F = 22^\circ (this is one of the acute angles)
  • GF=8GF = 8 (this is the length of one of the legs)
  • GG is the right angle, making HFHF the hypotenuse.

You are asked to find the following:

  • GHGH (the other leg of the triangle)
  • FGFG (this is already known as 8)
  • mHm\angle H (the other acute angle)

Step-by-step solution:

  1. Find GHGH using the tangent function:

    • We know that: tan(θ)=oppositeadjacent\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} For θ=22\theta = 22^\circ, the opposite side is GHGH and the adjacent side is FG=8FG = 8. So: tan(22)=GH8\tan(22^\circ) = \frac{GH}{8} Solving for GHGH: GH=8×tan(22)GH = 8 \times \tan(22^\circ) Using a calculator: GH8×0.4040=3.2GH \approx 8 \times 0.4040 = 3.2 So, GH3.2GH \approx 3.2.
  2. Find HFHF (the hypotenuse) using the cosine function:

    • The cosine of angle F=22F = 22^\circ is: cos(22)=GFHF\cos(22^\circ) = \frac{GF}{HF} So: HF=GFcos(22)=8cos(22)HF = \frac{GF}{\cos(22^\circ)} = \frac{8}{\cos(22^\circ)} Using a calculator: HF80.9272=8.63HF \approx \frac{8}{0.9272} = 8.63 So, HF8.6HF \approx 8.6.
  3. Find mHm\angle H: Since the sum of the angles in a triangle is always 180180^\circ, and we know one angle is 9090^\circ and F=22\angle F = 22^\circ, we can calculate H\angle H: mH=9022=68m\angle H = 90^\circ - 22^\circ = 68^\circ

Final answers:

  • GH3.2GH \approx 3.2
  • HF8.6HF \approx 8.6
  • mH=68m\angle H = 68^\circ

Would you like further details or have any other questions?

Here are 5 questions you could explore based on this problem:

  1. How would the solution change if the angle was 3030^\circ instead of 2222^\circ?
  2. Can you derive the same result using the Pythagorean theorem after finding one leg and the hypotenuse?
  3. How would you calculate the area of triangle HGF\triangle HGF?
  4. How would you calculate the perimeter of this triangle?
  5. What is the sine of F\angle F in this triangle, and how can it be used to verify the hypotenuse?

Tip: Always double-check your trigonometric ratios when using a calculator to avoid rounding errors.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Trigonometry
Right Triangle
Angles
Sine, Cosine, Tangent

Formulas

tan(θ) = opposite / adjacent
cos(θ) = adjacent / hypotenuse
sin(θ) = opposite / hypotenuse
Sum of angles in a triangle = 180°

Theorems

Pythagorean Theorem
Trigonometric Ratios

Suitable Grade Level

Grades 9-11