91Ó°ÊÓ

(a) The operation of "squaring" is a function: it takes a single real number \(x\) as input, and delivers a definite real number \(x^{2}\) as output. \- Every positive number arises as an output ("is the square of something" ). \(-\) Since \(x^{2}=(-x)^{2},\) each output (other than 0 ) arises from at least two different inputs. \- If \(a^{2}=b^{2},\) then \(0=a^{2}-b^{2}=(a-b)(a+b)\), so either \(a=b\), or \(a=-b\). Hence no two positive inputs have the same square, so each output (other than 0 ) arises from exactly two inputs (one positive and one negative). \- Hence each positive output \(y\) corresponds to just one positive input, called \(\sqrt{y}\). Find: (i) \(\sqrt{49}\) (ii) \(\sqrt{144}\) (iii) \(\sqrt{441}\) (iv) \(\sqrt{169}\) (v) \(\sqrt{196}\) (vi) \(\sqrt{961}\) (vii) \(\sqrt{96100}\) (b) Let \(a>0\) and \(b>0\). Then \(\sqrt{a b}>0\), and \(\sqrt{a} \times \sqrt{b}>0\), so both expressions are positive. Moreover, they have the same square, since $$ (\sqrt{a b})^{2}=a b=(\sqrt{a})^{2} \cdot(\sqrt{b})^{2}=(\sqrt{a} \times \sqrt{b})^{2} $$ \(\therefore \sqrt{a \times b}=\sqrt{a} \times \sqrt{b}\) Use this fact to simplify the following: (i) \(\sqrt{8}\) (ii) \(\sqrt{12}\) (iii) \(\sqrt{50}\) (iv) \(\sqrt{147}\) (v) \(\sqrt{288}\) (vi) \(\sqrt{882}\) (c) [This part requires some written calculation.] Exact expressions involving square roots occur in many parts of elementary mathematics. We focus here on just one example - namely the regular pentagon. Suppose that a regular pentagon \(A B C D E\) has sides of length \(1 .\) (i) Prove that the diagonal \(A C\) is parallel to the side \(E D\). (ii) If \(A C\) and \(B D\) meet at \(X,\) explain why \(A X D E\) is a rhombus. (iii) Prove that triangles \(A D X\) and \(C B X\) are similar. (iv) If \(A C\) has length \(x\), set up an equation and find the exact value of \(x\). (v) Find the exact length of \(B X\). (vi) Prove that triangles \(A B D\) and \(B X A\) are similar. (vii) Find the exact values of \(\cos 36^{\circ}, \cos 72^{\circ}\). (viii) Find the exact values of \(\sin 36^{\circ}, \sin 72^{\circ}\).

Step by step solution

01

Understanding and Solving Part (a)

To find the square roots in part (a), identify the positive number whose square equals the given value.(i) \(\sqrt{49} = 7\) as \(7^2 = 49\).(ii) \(\sqrt{144} = 12\) as \(12^2 = 144\).(iii) \(\sqrt{441} = 21\) as \(21^2 = 441\).(iv) \(\sqrt{169} = 13\) as \(13^2 = 169\).(v) \(\sqrt{196} = 14\) as \(14^2 = 196\).(vi) \(\sqrt{961} = 31\) as \(31^2 = 961\).(vii) \(\sqrt{96100} = 310\) as \(310^2 = 96100\).

02

Using the Simplification Rule in Part (b)

Leverage the property \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\) to simplify the following.(i) \(\sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2}\).(ii) \(\sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3}\).(iii) \(\sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2} = 5\sqrt{2}\).(iv) \(\sqrt{147} = \sqrt{49 \times 3} = \sqrt{49} \times \sqrt{3} = 7\sqrt{3}\).(v) \(\sqrt{288} = \sqrt{144 \times 2} = \sqrt{144} \times \sqrt{2} = 12\sqrt{2}\).(vi) \(\sqrt{882} = \sqrt{81 \times 2 \times 3^2} = 9 \times 3\sqrt{2} = 27\sqrt{2}\).

03

Solving Part (c)(i) - Proving Parallel Lines in a Pentagon

For a regular pentagon, all internal angles are 108°, and the pentagon is symmetric. Therefore, line segment AC is horizontal and parallel to line segment ED based on their positions and the symmetry of the pentagon.

04

Understanding Part (c)(ii) - Prove Quadrilateral is a Rhombus

In a symmetric figure like the regular pentagon, if AC and BD intersect at X, triangle AXD has equal-length sides AX and DX because of symmetry. Thus, combining with AB=AE=1, quadrilateral AXDE is a rhombus.

05

Solving Part (c)(iii) - Proving Similarity of Triangles

Since triangles ADX and CBX share angle DXA=BCX and both have AX=BX (since they are parts of the same diagonal), triangles ADX and CBX are similar by the AA criterion.

06

Solving Part (c)(iv) - Find Exact Value of x

Given the symmetry and equal diagonals in a pentagon, the relation \(x^2 - 4x + 1 = 0\) forms. Solving this quadratic gives \(x = 2 + \sqrt{5}\) or \(x = 2 - \sqrt{5}\); the former is possible given the situation.

07

Solving Part (c)(v) - Length of BX

Using triangle similarity, \(BX = AD - AX\). Since AX is half of the diagonal AC, \(BX = \sqrt{5}-1\) when \(AC=x=2+\sqrt{5}\).

08

Solving Part (c)(vi) - Proving Similarity in Another Triangle Pair

Triangles ABD and BXA are similar because they both have angle BXA = ABD and angle BAD = BAX.

09

Solving Part (c)(vii) - Exact Values of Cosines

Using the isosceles triangle properties and \(x = 2+\sqrt{5}\), formulas yield \(\cos 36^{\circ} = \frac{1+\sqrt{5}}{4}\) and \(\cos 72^{\circ} = \frac{\sqrt{10-2\sqrt{5}}}{4}\).

10

Solving Part (c)(viii) - Exact Values of Sines

Using Pythagorean identity, \(\sin 36^{\circ} = \frac{\sqrt{10-2\sqrt{5}}}{4}\) and \(\sin 72^\circ = \frac{1+\sqrt{5}}{4}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Real Numbers

Real numbers consist of all numbers, including those that are rational (such as integers and fractions) and those that are irrational (like the square roots of non-perfect squares). The operation of squaring a real number is significant because it always results in a positive number or zero. This means for any real number \( x \), squaring it will yield \( x^2 \), which is always a non-negative real number.

Understanding real numbers is fundamental because they form the building blocks of many mathematical concepts, including roots. If you know that \( a^2 = b^2 \), it implies that \( a = b \) or \( a = -b \). This property is crucial when finding square roots since a positive real number output corresponds to one positive and one negative input. These inputs illustrate how every real number has its place within the number line continuum.

Symmetric Figures

Symmetry in mathematics often implies that one half of a figure is a mirror image of the other. Symmetric figures are vital in geometry and make problems easier to solve due to their predictable nature.

For instance, consider a regular pentagon. This is a symmetric figure that has equal side lengths and internal angles of 108°. Symmetry allows us to determine relationships between different parts of the figure without computing each individually. For instance, in a regular pentagon, diagonals like AC might be parallel to sides like ED, stemming from its symmetry. Such patterns simplify proving geometric relationships, like parallelism and congruence, that are inherent within these shapes.

Therefore, understanding symmetry can greatly ease solving complex shapes by identifying these inherent and consistent properties.

Similar Triangles

Triangles are said to be similar if they have the same shape, but not necessarily the same size. This means their corresponding angles are equal, and their corresponding sides are proportional.

In a geometric setting, similar triangles are incredibly useful because knowing just a few elements about one triangle can inform us about the other. Consider two triangles ADX and CBX within a pentagon. If they share an angle and have proportional adjoining sides, they are similar by the Angle-Angle criterion (AA).

Using similarity allows us to make deductions about side lengths and angles indirectly, reducing complex calculations. For instance, if you know the lengths of sides in one triangle, you can find unknown sides in a similar triangle by setting up a proportion based on similarity. This is a powerful concept when dealing with complex geometric problems.

Trigonometric Values

Trigonometric values are derived from the angles of a triangle, primarily focusing on the relationships between their sides. They are typically expressed with functions like sine (sin) and cosine (cos).

For special angles like 36° and 72°, trigonometric formulas provide exact values rather than decimal approximations. These values are important in problems involving isosceles triangles, such as those present in pentagons. For example, the exact value of \( \cos 36^{\circ} \) is \( \frac{1+\sqrt{5}}{4} \), and \( \cos 72^{\circ} \) is \( \frac{\sqrt{10-2\sqrt{5}}}{4} \). Similarly, their sine counterparts reflect these relationships with the Pythagorean identity.

Understanding these values helps solve trigonometric problems more accurately and efficiently. It especially aids in providing precise solutions in geometrical constructions and proofs, where decimal approximations might fall short.