Question

In a right angled triangle, if the hypotenuse is four times as long as the perpendicular from the opposite vertex, one of the acute angles is (a) \(60^{\circ}\) (b) \(45^{\circ}\) (c) \(30^{\circ}\) (d) \(15^{\circ}\)

Short answer

Question: In a right-angled triangle, the hypotenuse is 4 times the length of the perpendicular from the opposite vertex. Determine the value of one of the acute angles in the triangle: (a) \(10^{\circ}\) (b) \(12^{\circ}\) (c) \(14^{\circ}\) (d) \(15^{\circ}\) Answer: (d) \(15^{\circ}\) (approximately)

Step-by-step solution

Set up the given information

The hypotenuse, denote it by \(h\), is four times the length of the perpendicular from the opposite vertex. Let the length of this perpendicular be \(p\). So we have \(h = 4p\). Let's denote the remaining side length as \(b\). Now let's denote one of the acute angles as \(A\) and the other as \(B\).

Use trigonometric ratios to find possible relationships between the angles and sides

We know that in a right-angled triangle, sin(A) = \(\frac{opposite}{hypotenuse}\) sin(B) = \(\frac{opposite}{hypotenuse}\) cos(A) = \(\frac{adjacent}{hypotenuse}\) cos(B) = \(\frac{adjacent}{hypotenuse}\) tan(A) = \(\frac{opposite}{adjacent}\) tan(B) = \(\frac{opposite}{adjacent}\) In our case, we can use the tan() function because we know the ratio of the hypotenuse and the perpendicular side lengths. From the triangle, tan(A) = \(\frac{p}{b}\) tan(B) = \(\frac{b}{p}\)

Set up the equation using given information in the triangle

We know the hypotenuse is four times the perpendicular length. If we look at the tan(A) formula: tan(A) = \(\frac{p}{4p - p} = \frac{p}{3p}\) We can simplify this to: tan(A) = \(\frac{1}{3}\)

Determine the angle A using inverse tangent

Now, we can use the inverse tangent function to find the value of angle A: \(A = tan^{-1}(\frac{1}{3})\) which approximately equals 18.43 degrees. Now we see that none of the given options are correct, but the closest answer would be (d) \(15^{\circ}\).

Key concepts

Trigonometric Ratios

Trigonometric ratios are fundamental in the field of geometry, particularly when dealing with right-angled triangles. These ratios relate the angles of a triangle to the lengths of its sides, serving as a bridge between angle measures and side lengths in right triangles.

For any angle in a right-angled triangle, there are three basic trigonometric ratios:

• The sine (sin) function compares the length of the side opposite to the angle with the hypotenuse.
• The cosine (cos) function compares the length of the adjacent side to the angle with the hypotenuse.
• The tangent (tan) function compares the length of the side opposite to the angle with the side adjacent to the angle.

Using these ratios, anyone can determine the measure of an angle if the lengths of two sides of a triangle are known, or vice versa. It's crucial to remember these ratios as they are used to solve many problems in geometry.

Inverse Tangent Function

When you're given the tangent of an angle and need to find the angle itself, you turn to the inverse tangent function, often denoted as \(tan^{-1}\) or arctan. This function essentially works backwards from the tangent function – instead of finding the ratio of the opposite to adjacent side lengths, you're finding the measure of the angle that produces this ratio.

It's important to distinguish between the tangent and the inverse tangent functions. While the tangent of an angle gives you a ratio (a real number), the inverse tangent of a ratio gives you an angle (measured in degrees or radians). This is particularly useful in various fields, including physics, engineering, and of course, mathematics itself. In exercises that involve right-angled triangles, if two side lengths are known, the inverse tangent function allows you to find the angles easily.

Geometry in Mathematics

Geometry is a vast area of mathematics that deals with shapes, sizes, the properties of space, and the relationships between figures and spaces. Right-angled triangles are one of the fundamental shapes studied in geometry because they form the basis of trigonometry.

The rules and principles applied within geometry, such as those involving angles and sides of triangles, are not only theoretical but also have practical applications in numerous fields such as architecture, design, astronomy, and even art. Understanding these concepts and being able to apply them to solve problems is a key skill in mathematics.