The sides of a triangle are 6, 7, and x. What is the largest value that could be the area of the triangle?

Question

Is it a right triangle? And will it always be a right triangle?

Answer 1

Area of any triangle with sides of lengths a, b, c is

A = 0.25*√[(a + b + c)(a + b - c)(b + c - a)(c + a - b)]

Here a = 6, b = 7, c = x.

For 'A' to be maximum, [(a + b + c)(a + b - c)(b + c - a)(c + a - b)] has to be maximum.

Let y = (a + b + c)(a + b - c)(b + c - a)(c + a - b).
Here, y = (6 +7 + x)(6 +7 - x)(7 + x - 6)(x + 6 - 7)
or y = (13 + x)(13 - x)(x + 1)(x - 1) = (169 - x^2)(x^2 - 1)
or y = -169 + 170x^2 - x^4
or y = -169 + 85^2 - (85^2 - 2*85x^2 + x^4)
or y = 7056 - (85 - x^2)^2

As (85 - x^2)^2 is a real square so it cannot be negative.
So less is its value, more is the value of 'y'.
Minimum value of (85 - x^2)^2 is zero.
Clearly, 'y' is maximum when x^2 = 85,
So, the largest value of the area of the triangle is
0.25*√y = 0.25*√7056 = 21 sq. units.

Answer 2

Let x be the base. Then, if 7 is the heigh, 6 cant be the hypothenuse.

If 7 is the base, and x is the heigh, then the hypothenuse will be 6. But, if 6 is the heigh and x the hypothenuse, then you will have the largest value for the area

6*7/2 = 21 unit^2
Ana

Answer 3

The triangle will have the largest possible area if it is
right angled.

So x will be the hypotenuse or x= Sq rt of (6^2 + 7^2)

and the area is (6^7)/2 or 21

Answer 4

Let a be the angle between those two sides.

The area is given by:

A = 1/2*6*7*Sin(a) = 21*Sin(a)

The maximum value is when Sin(a) has the maximum value, therefore when Sin(a) = 1. The largest value is:

A = 21*Sin(pi/2) = 21

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You can also determine it by evaluating when the first derivative of that function equals zero:

A' = 21*Cos(a)

21*Cos(a) = 0

Cos(a) = 0

a = pi/2 (+ k*pi)

Answer 5

the largest area will always be a right triangle

Answer 6

i think it would be a right triangle since the height is highest.
(7*6)/2 = 21

Answer 7

the area of a triangle is base times height divided by 2.
asquared=bsquared=csquared so 36+49=85 the square root is approx 9. so 9*7=63 /2= 31.5 the largest value that could be the area of the traingle is 31.5.

Answer 8

The area could be infinitely big because x could be infinitely long. The info you provided in your question does not indicate that it must be a right triangle, although it can be.

Answer 9

Given the sides 6 and 7, you know that the greatest integer side can only be 12, because the sum of two sides must be greater than the other side

You know that it is not a right triangle, because pythagorean's theorem is not satisfied. 6^2+7^2 is not 12^2.

using Heron's formula, you can find the area
s = (a+b+c)/2
A = sqrt(s(s-a)(s-b)(s-c))

s=(6+7+12)/2 = 12.5
A = sqrt(12.5(12.5-6)(12.5-7)(12.5-12))
A=14.948

Answer 10

7*6/2= 21

Yes,Yes