Problem 23 - Algebra 2

Question

In \(\triangle A B C, \angle C\) is a right angle. Find the remaining sides and angles. Round your answers to the nearest tenth. \(a=17, c=22\)

Step-by-Step Solution

Answer

The remaining side (b) is approximately 12.37 units long. The remaining angles (A and B) are approximately 54.0° and 36.0° respectively.

Step 1: Finding the length of side b

We can use the Pythagorean theorem to calculate the length of side b. The Pythagorean theorem is \(a^2 + b^2 = c^2\), where c is the hypotenuse, while a and b are the lengths of the other two sides. Rearrange the equation to solve for b: \(b^2 = c^2 - a^2\). Subsitute the given values (a = 17, c = 22) into the equation. \(b^2 = 22^2 - 17^2\). Evaluating this, we get \(b \approx 12.37\).

Step 2: Calculating angle A

The triangle is right-angled atAngle C, so we can find the other two angles using the inverse trigonometric functions. Use the tangent ratio to find angle A, because we know the lengths of the sides adjacent and opposite to angle A. The tangent of an angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. Thus, \(\tan(A) = a/b\). Using the inverse tangent function or arctan, we find \(A = \arctan(a/b) = \arctan(17/12.37)\). Evaluate this to get \(A \approx 54.0^\circ\).

Step 3: Calculating angle B

In any triangle, the sum of the angles is 180 degrees. Since we already know two of the three angles (one is right angle, 90°, and the other, A, we just calculated), we can subtract these two from 180 to find the third angle. So, \(B = 180^\circ - 90^\circ - A = 180^\circ - 90^\circ - 54.0^\circ\). Evaluate this to get \(B \approx 36.0^\circ\).

Key Concepts

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry, especially concerning right triangles. It states that in a right triangle, the square of the length of the hypotenuse (which is the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This can be expressed with the formula:

• \[ a^2 + b^2 = c^2 \]

In the exercise provided, we know two sides of the triangle: side \(a = 17\) and the hypotenuse \(c = 22\). Utilizing the theorem allows us to find the third side, \(b\). We rearrange the formula to:

• \[ b^2 = c^2 - a^2 \]

By substituting the known values, we calculate:

• \[ b^2 = 22^2 - 17^2 \]
• \[ b^2 = 484 - 289 \]
• \[ b^2 = 195 \]
• \[ b \approx \sqrt{195} \approx 13.96 \]

Thus, side \(b\) is approximately 13.96 when rounded to the nearest tenth.

Trigonometric Functions

Trigonometric functions relate the angles and sides of right triangles. They include sine, cosine, and tangent, which are pivotal in solving triangles:

• Tangent is defined as the ratio of the opposite side to the adjacent side of a given angle in a right triangle.
• Formulaically:
  • \[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]

In our exercise, we use the tangent function to find angle \(A\) because we know the opposite (side \(a = 17\)) and the adjacent (side \(b \approx 13.96\)). To determine \(A\), we apply the arctangent, or inverse tangent function:

• \[ A = \arctan\left(\frac{a}{b}\right) = \arctan\left(\frac{17}{13.96}\right) \]
• \[ A \approx 50.7^\circ \]

Here, we round to the nearest tenth, thus confirming angle \(A\) is approximately 50.7 degrees.

Triangle Angles Sum

A crucial rule in geometry is that all angles in a triangle add up to 180 degrees. This rule is especially useful when you know two of the angles in a triangle and need to find the third.

• In our scenario, \(\triangle ABC\) is a right triangle, so one angle, angle \(C\), is automatically \(90^\circ\).
• With angle \(A\) found to be approximately \(50.7^\circ\), we can calculate the remaining angle \(B\) by subtracting the known angles from \(180^\circ\):
  • \[ B = 180^\circ - 90^\circ - 50.7^\circ \]
  • \[ B \approx 39.3^\circ \]

Thus, angle \(B\) is approximately 39.3 degrees, rounding to the nearest tenth finalizes our solution.