Problem 36 - Algebra 2

Question

In \(\triangle A B C, \angle C\) is a right angle. Two measures are given. Find the remaining sides and angles. Round your answers to the nearest tenth. \(a=7, b=10\)

Step-by-Step Solution

Answer

The hypotenuse \(c\) of the triangle is approximately 12.2 units. The measure of \(\angle A\) is approximately 44.4° and measure of \(\angle B\) is approximately 45.6°.

Step 1: Find the Hypotenuse

Using the Pythagorean theorem, which states that \(a^2 + b^2 = c^2\) in all right angled triangles, here \(a = 7\) and \(b = 10\). After substituting the values, compute length of the hypotenuse \(c\).

Step 2: Calculate \(\angle A\)

Use the trigonometric relation of sine which is defined as the ratio of the opposite side to the hypotenuse in a right angled triangle. Apply the formula \(\sin A = a/c\). By taking the arcsin or inverse sine on both sides, the measure of \(\angle A\) can be calculated.

Step 3: Calculate \(\angle B\)

Since the sum of angles in a triangle is always 180°, and in a right angled triangle one of the angles is 90°, \(\angle B\) can be calculated as \(\angle B = 180° - 90° - \angle A\).

Key Concepts

Pythagorean theorem

The Pythagorean theorem is a fundamental principle in geometry specifically applied to right triangles. It's a rule that relates the lengths of the sides. When you have a right triangle, one of the angles is exactly 90 degrees. This creates a unique relationship between the triangle's sides. If you label the sides of the triangle as \(a\), \(b\), and \(c\), where \(c\) is the hypotenuse (the side opposite the right angle), the theorem states:

• \(a^2 + b^2 = c^2\)

For example, if you know the lengths of sides \(a\) and \(b\) (7 and 10 in this problem), you can find the hypotenuse by plugging these values into the equation. Calculate \(c\) by \(c = \sqrt{a^2 + b^2}\), which gives you the length of the hypotenuse. This powerful theorem is key to finding unknown measurements in right triangles.

right triangle

A right triangle is a special type of triangle where one of the angles is exactly 90 degrees. This unique characteristic allows us to apply specific mathematical properties and formulas that are not valid for other triangles. When examining a right triangle, you can easily identify several parts:

• Hypotenuse: The longest side, opposite the right angle.
• Base and Height: The other two sides that form the right angle.

The distinct property of a right triangle simplifies calculation of its elements. By using trigonometric functions and the Pythagorean theorem, you can solve for unknown sides and angles. This makes right triangles extremely practical in various fields like physics, engineering, and architecture.

sine function

The sine function is one of the three primary trigonometric functions and it is especially useful in the context of right triangles. In a right triangle, the sine of an angle (other than the right angle) is defined as the ratio of the length of the opposite side to the length of the hypotenuse. This can be mathematically described by the formula:

• \( \sin A = \frac{\text{opposite}}{\text{hypotenuse}} \)

Here, the angle \(A\) is one of the non-right angles. In our example, this would mean \( \sin A = \frac{7}{c} \), where \(7\) is the side opposite angle \(A\) and \(c\) is the hypotenuse. By solving this equation for \(A\) using the inverse sine function \( \sin^{-1} \), you can find the specific measure of angle \(A\). This process helps derive unknown angles when side lengths are given.

angle calculation

Calculating angles in a right triangle can be straightforward once one angle or side measurement is known. We've seen how to use the sine function to find one angle, but what about the other? In a right triangle, the sum of the angles must equal 180 degrees. Since we know one of these angles is 90 degrees, we can use this principle to find the remaining angle. If you know angle \(A\) using trigonometry:

• Use \(\angle B = 180° - 90° - \angle A\)

This is simply because the total angle measure of a triangle is fixed at 180 degrees. Often, finding one angle directly gives you enough information to calculate the other, making this process fast and efficient. Understanding these calculations becomes handy, particularly in solving geometry problems and real-world applications.