Math Problem Statement

In a right-angle triangle, ∠B = 90°, BC = 3 cm, and AC = 5 cm. Find the length of the remaining side and the angles of the triangle.

Solution

This problem involves a right-angled triangle where the angle B=90B = 90^\circ, the length of side BC=3BC = 3 cm, and the hypotenuse AC=5AC = 5 cm. We need to find the remaining side and the other angles of the triangle.

Solution:

Given:

  • B=90\angle B = 90^\circ
  • BC=3BC = 3 cm (one of the legs)
  • AC=5AC = 5 cm (the hypotenuse)

Using the Pythagorean theorem: AC2=AB2+BC2AC^2 = AB^2 + BC^2 Plugging in the known values: 52=AB2+325^2 = AB^2 + 3^2 25=AB2+925 = AB^2 + 9 Subtract 9 from both sides: 16=AB216 = AB^2 Now, take the square root of both sides: AB=4cmAB = 4 \, \text{cm}

Finding the Angles:

Since it's a right-angled triangle, we already have B=90\angle B = 90^\circ. We now need to find the other two angles, A\angle A and C\angle C.

Using trigonometry, we can find these angles. Let's start with A\angle A: sinA=oppositehypotenuse=BCAC=35\sin A = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{BC}{AC} = \frac{3}{5} A=arcsin(35)36.87A = \arcsin\left(\frac{3}{5}\right) \approx 36.87^\circ

Since the sum of angles in a triangle is 180180^\circ: C=1809036.87=53.13C = 180^\circ - 90^\circ - 36.87^\circ = 53.13^\circ

Final Answer:

  • The length of side AB=4AB = 4 cm.
  • A36.87\angle A \approx 36.87^\circ.
  • C53.13\angle C \approx 53.13^\circ.

Would you like more details or have any questions?

Further Questions:

  1. How would you solve this problem using the cosine or tangent functions?
  2. How can the sine rule be applied to find the angles of this triangle?
  3. What happens to the other angles if BCBC is increased to 4 cm?
  4. How does the Pythagorean theorem change if the triangle is not right-angled?
  5. Can you use the properties of special right triangles to simplify this problem?

Tip:

In right-angled triangles, knowing two sides or one side and one angle (other than the right angle) allows you to find all other sides and angles using basic trigonometric functions.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Trigonometry
Right-Angled Triangles
Pythagorean Theorem

Formulas

AC² = AB² + BC² (Pythagorean theorem)
sin(θ) = opposite / hypotenuse
cos(θ) = adjacent / hypotenuse
tan(θ) = opposite / adjacent

Theorems

Pythagorean Theorem
Basic Trigonometric Functions

Suitable Grade Level

Grades 7-10