Problem 21 - 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. \(a=8.1, b=6.2\)

Step-by-Step Solution

Answer

The length of side c is calculated as approximately \( 10.1 \) units. Angle A measures approximately \( 36.9^{\circ} \) and angle B measures approximately \( 53.1^{\circ} \).

Step 1: Calculate the Length of Side c

As \( \triangle ABC \) is a right-angled triangle at angle C, Pythagoras' theorem applies. The theorem states that in a right-angled triangle the square of the length of the hypotenuse (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 written as \( c = \sqrt{a^{2} + b^{2}} \). Here, \( a = 8.1 \) and \( b = 6.2 \). By substituting the values of a and b in the formula we find the length of side c.

Step 2: Calculate Angle A

Knowing all three sides of the triangle, we can calculate angle A using the tangent ratio. The tangent of an angle in a right angled triangle is the ratio of the length of the opposite side to the length of the adjacent side. So, \( \tan(A) = \frac{{b}}{{a}} \). Therefore, to find angle A we can use the inverse of the tangent function, known as arctangent or atan. We write it as \( A = \arctan\left(\frac{{b}}{{a}}\right) \). By substituting \( b = 6.2 \) and \( a = 8.1 \) we get the measurement of angle A.

Step 3: Calculate Angle B

Now, in any triangle, the sum of the angles is always \( 180^{\circ} \). So we could find the remaining angle by subtracting the known angles from \( 180^{\circ} \). Written as a formula, this is \( B = 180^{\circ} - A - 90^{\circ} \). By substituting the value of angle A found previously, we get the measurement of angle B.

Key Concepts

Pythagorean Theorem

The Pythagorean Theorem is a cornerstone of geometry, especially when working with right-angled triangles. It helps us determine the length of the hypotenuse, which is the longest side, opposite the right angle. For triangle \(\triangle ABC\), where \(\angle C\) is a right angle, the theorem is written as \(c^2 = a^2 + b^2\). Here, the sides \(a\) and \(b\) are the shorter sides that form the right angle.

To use the Pythagorean Theorem, square the lengths of \(a\) and \(b\), add these squares together, and then take the square root of the result to find \(c\). In our example, with \(a = 8.1\) and \(b = 6.2\):

• First, calculate \(a^2 = 8.1^2 = 65.61\)
• Then, calculate \(b^2 = 6.2^2 = 38.44\)
• Add them: \(65.61 + 38.44 = 104.05\)
• Finally, \(c = \sqrt{104.05} = 10.2\), rounded to the nearest tenth.

These steps reveal why the hypotenuse is critical: it's the key to solving many problems involving right triangles.

Angle Calculation

Finding angles in right-angled triangles often involves using trigonometric ratios. To find angle \(A\) in \(\triangle ABC\) where \(\angle C\) is 90°, we use the tangent ratio. Tangent relates the opposite side and adjacent side of an angle through the formula \(\tan(A) = \frac{b}{a}\).

Given \(b = 6.2\) and \(a = 8.1\), we substitute these values to get \(\tan(A) = \frac{6.2}{8.1}\):

• Compute the division: \(\tan(A) = 0.765\)
• Using a calculator, find \(A = \arctan(0.765)\), which gives \(A = 37.5^{\circ}\).

We've calculated angle \(A\), so now to find angle \(B\). In any triangle, all angles add up to \(180^{\circ}\). Since \(\angle C = 90^{\circ}\), then \(B = 180^{\circ} - 90^{\circ} - 37.5^{\circ} = 52.5^{\circ}\).

By understanding these relationships, you can determine all angles in a right triangle using just one angle measure.

Trigonometric Ratios

Trigonometric ratios are a powerful tool in examining right triangles. They originate from the ancient Greeks who needed methods to calculate distances and angles. In a right triangle, there are three primary trigonometric ratios: sine, cosine, and tangent, each representing a different aspect of the triangle's sides relative to its angles.

• Sine (\(\sin\)) is the ratio of the opposite side to the hypotenuse.
• Cosine (\(\cos\)) involves the adjacent side and the hypotenuse.
• Tangent (\(\tan\)) compares the opposite to the adjacent side.

These ratios can be visualized as part of a triangle's internal structure, useful for calculating unknowns when some values are given. In our example problem, tangent is used to find angle \(A\), since it directly relates the opposite side \(b\) and the adjacent side \(a\).

The trigonometric ratios also play a role in real-world problems—from construction to aircraft navigation—where calculating angles and distances with precision is essential. Understanding how \(\tan(A)\) finds angle \(A\) was a single step, but each ratio has unique applications. Through practice with these concepts, you'll gain confidence in solving complex problems involving right triangles.