How do you find the base of a triangle

Definition

Base meaning bottom, it refers to any side of a triangle, which is perpendicular to its height or altitude. For scalene or equilateral triangle, any side can be the base, while for an isosceles triangle it is the single, unequal side.

Formula

The formula for base of a triangle can be derived from the standard formula of area of a triangle as shown below:

As we know,

Area (A) = ½ (b x h), here b = base, h = height

=> 2A = b x h

=> b = 2A/h

Hence, mathematically, base of a triangle can also be defined as twice the area divided by the height of the triangle.

How to Find the Base of a Triangle

Let us solve some problems to derive the base of the given triangles.

Problem: Finding the base of a triangle, when the HEIGHT and the AREA are known.

Find the base of a triangle with area 12 cm2 and height 8 cm.

Solution:

As we know,
b = 2A/h, here area = 12cm2, h = 8 cm
= 2 x 12/8
= 3 cm

Problem: Finding the base of a right triangle, when the HEIGHT and the HYPOTENUSE are known.

Identify the base of a right triangle having height 6 cm and hypotenuse 9 cm.

Solution:

Here, we will use the Pythagorean Theorem,
(Hypotenuse)2 = (Base)2 + (Height)2, here height = 6 cm and hypotenuse = 9 cm
(6) 2 = (b)2 + (3)2
(b)2 = (6) 2 – (3)2
b2 = 36- 9
b = √25 = 5 cm.

Problem: Finding the base of an isosceles triangle, when the SIDES and the HEIGHT are known.

Solution:

Here, we will use the Pythagorean Theorem,
In △ABC,
Since, the height AE divides the △ABC into two similar right triangles, △ABE and △ACE and the base BC into BE and EC equally
Such that,
BC = BE + EC
Now,
 According to the Pythagoras theorem,
(Hypotenuse)2 = (Base)2 + (Height)2
 In △ABE,
(AB)2 = (BE)2 + (AE)2, here AB = 3 cm, AE = 5 cm
(BE)2= (AB)2 – (AE)2
(BE)2 = (5) 2 – (3)2
(BE)2 = 25 – 9
(BE)2 = 16
BE = 4 cm
Similarly, EC = 4 cm
Thus,
BC = BE + EC = 8 cm

Solution:

Let the base = 2x and height = h
As we know,
Area of △ABC = ½ (b x h) = 60 cm2
⇒½(b x h) = 60, here b = 2x
⇒ ½ x 2x x h = 60
⇒ xh = 60 …… (1)
Now, using Pythagorean theorem in △ABD, we get
AB2 = AD2 + BD2
132 = h2 + x2
⇒ h2 + x2 = 169
Adding 2hx on both sides we get,
⇒ h2 + x2 + 2hx = 169 + 2hx
⇒ (h + x)2 = 169 + 2 x 60
⇒ (h + x)2 = 289
⇒ h + x = 17
⇒ h = 17 – x …… (2)
Substituting (2) in (1), we get,
x (17 – x) = 60
17x – x2 = 60
⇒ -x2 + 17x – 60 = 0
⇒ x2– 17x + 60 = 0
⇒ x2– 12x – 5x +60 = 0
⇒ x(x-12) – 5(x + 60)  =0
⇒ (x-12) (x – 5) = 0
Either,
 x -12 = 0
x = 12
Or,
x – 5 =0
x = 5
Hence, base of the given isosceles triangle is,
b = 2x = 2 x 5 =10 cm (if x = 5) or, 2x = 2 x 12 = 24 cm (if x = 12)

The base is 6 in.

The formula for the area of a triangle is

#"Area "= 1/2 × "base × height "# or

#A = 1/2bh#

We know #h# and #A#, so let's rearrange the formula and solve for A.

#2A = bh# or #bh = 2A#

#b = (2A)/h = ("2 × 48 in"^cancel(2))/(16 cancel("in")) = "6 in"#

Check:

#A = 1/2bh = 1/2 × "6 in × 16 in" = "48 in"^2#

The Pythagorean Theorem, an equation that shows the relationship between a right triangle's three sides, can help you to find the length of its base. A triangle that contains a 90-degree or right angle in one of its three corners is called a right triangle. A right triangle's base is one of the sides that adjoins the 90-degree angle.

TL;DR (Too Long; Didn't Read)

The Pythagorean Theorem is essentially, ​a​2 + ​b​2 = ​c​2. Add side ​a​ times itself to side ​b​ times itself to arrive at the length of the hypotenuse, or side ​c​ times itself.

The Pythagorean Theorem

The Pythagorean Theorem is a formula that gives the relationship between the lengths of a right triangle's three sides. The triangle's two legs, the base and height, intersect the triangle's right angle. The hypotenuse is the side of the triangle opposite the right angle. In the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:

a^2 + b^2 = c^2

In this formula, ​​a​​ and ​​b​​ are the lengths of the two legs and ​​c​​ is the length of the hypotenuse. The ​2​ signifies that ​​a​​, ​​b​​, and ​​c​​ are ​squared​. A number squared is equal to that number multiplied by itself – for example, 42 is equal to 4 times 4, or 16.

Finding the Base

Using the Pythagorean theorem, you can find the base, ​a​, of a right triangle if you know the lengths of the height, ​b​, and the hypotenuse, ​c​. Since the hypotenuse squared is equal to the height squared plus the base squared, then:

a^2 = c^2 - b^2

For a triangle with a hypotenuse of 5 inches and a height of 3 inches, find the base squared:

c^2 - b^2 = (5 × 5) - (3 × 3) = 25 - 9 = 16 \\ \implies a = 4

Since b2 equals 9 , then ​a​ equals the number that, when squared, makes 16. When you multiply 4 by 4, you get 16, so the square root of 16 is 4. The triangle has a base that is 4 inches long.

A Man Called Pythagoras

The Greek philosopher and mathematician, Pythagoras, or one of his disciples, is attributed with the discovery of the mathematical theorem still used today to calculate the dimensions of a right triangle. To complete the calculations, you must know the dimensions of the longest side of the geometric shape, the hypotenuse, as well as another one of its sides.

Pythagoras migrated to Italy in about 532 BCE because of the political climate in his own country. Besides being credited with this theorem, Pythagoras – or one of the members of his brotherhood – also determined the significance of numbers in music. None of his writings have survived, which is why scholars don't know if it was Pythagoras himself who discovered the theorem or one of the many students or disciples who were members of the Pythagorean brotherhood, a religious or mystical group whose principles influenced the work of Plato and Aristotle.

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