Answer to Question #158205 in Geometry for Janine Leodones

Question #158205

Express:

a.

the altitude h of an equilateral triangle as a function of its side s

b.

the hypotenuse c of a r

ight triangle as a function of side s.

c.

a side s of a square as a function of its diagonal d.

d.

the median m of a trapezoid as a function of its area A and its altitude h.



1
Expert's answer
2021-02-16T14:03:21-0500

a)

Equilateral triangles have sides of equal length, with angles of 60°. To find the height, we can draw an altitude to one of the sides in order to split the triangle into two equal 30-60-90 triangles.

Now, the side of the original equilateral triangle (lets call it "s") is the hypotenuse of the 30-60-90 triangle. Because the 30-60-90 triangle is a special triangle, we know that the sides are "x" ,"x\\sqrt3" and "2x" , respectively.

Thus, s = 2x and x = s/2.

now, in right angled triangle: by pythagoras theorem:

"hypotenuse^2=perpendicular^2+base^2"

"(2x)^2=height^2+x^2"

"[2*(\\frac{s}{2})]^2=h^2+[\\frac{s}{2}]^2"

"h=\\frac{s\\sqrt 3}{2}"

Height of the equilateral triangle: "h=\\frac{s\\sqrt 3}{2}"


b)

in a right angled triangle, one angle is "90\\degree" and another angles are let as "\\alpha" and "\\beta", let angle "\\alpha" is between the base and hypotenuse and angle "\\beta" is between the hypotenuse (c) and perpendicular then, by spliting the hypotenuse (c) in sine and cosine components with base angle "\\alpha" are as:

base component is = "c*cos\\alpha"

perpendicular component = "c* sin\\alpha"

let the side (s) is perpendicular then : "s=c*sin\\alpha"

or hypotenuse (c) = "\\frac{s}{sin\\alpha}"

or let side (s) is base of triangle then : "s=c*cos\\alpha"

or hypotenuse (c) = "\\frac{s}{cos\\alpha}"


c)

let a square ABCD (side s), let's join A and C to form a diagonal of the square, now in the square, a right angle triangle is form that is ADC right angled at D and the side AD and DC is equal to s and the diagonal AC (hypotenuse) is d. By pythagoras theorem :

"(AD)^2+(DC)^2=(AC)^2"

s2+s2 = d2

2s2 = d2

"s=\\frac{d}{\\sqrt2}"


d)

The median (also called a midline or midsegment) is a line segment half-way between the two bases or parallel lines.

The median's length is the average of the two base lengths:

m = (a+b)/2

where,

m= median's length

a and b are the lengths of parallel lines or the lengths of two bases.

We know that

area of trapezoid = average of both base length * height (h)

A = [(a+b)/2]*h

or A = m*h as m = (a+b)/2

thus median of trapezoid can be written as "m = \\frac{A}{h}"


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