Is the answer to this geometry question 21 inches???

Question

The height and sides of a triangle are four consecutive integers. The height is the first integer, and the base is the fourth integer. The perimeter of the triangle is 42 in. Find the area of the triangle.


If you got something other than 21 inches, can you explain how you got it??? Thanks!

Answer 1

The area should be 90 SQUARE INCHES

So three consecutive integers are the sides, which add up to 42. Obviously they are 13, 14, 15. So 15 is the base.
The height must then be 12.

So area = 1/2 * 12 * 15 = 90
It should be 6 x 15 not 6 + 15. That may have been your mistake.

Answer 2

*** THERE IS NO SOLUTION. After some additional thought, I have made a correction to my previous post. The correction is bookend end by '***'. Some of the calculations required here are identical to those that follow in the previous post, so I will not explain them at length.

It is easy to verify that the three consecutive numbers which give a sum of 42 are 13,14,15. This leads two sets of possible numbers: 12,13,14,15 or 13,14,15,16.

To choose between these numbers notice that the height must be either 16 or 12. However, with some thought, it is clear that 16 cannot be the height, for it would exceed the length of the longest side, and the height must be less than this side.

So the numbers we must use are 12,13,14,15. This is obviously a scalene triangle, but it cannot be a right triangle because the height is not equal to any of the sides. Therefore it is an oblique scalene triangle. Therefore we don't yet know the length of the base. We can however still find the area, and with knowledge of the height and area, we can calculate the measure of the base.

To find the area of a triangle without knowing anything about its angles or the measures of both the height and base, we can use Heron’s formula. In preparation for Heron's formula we need to calculate what is known as the semi perimeter, which we will designate 's'. We will name the sides 'a','b','c' as is convention:
a=13
b=14
c=15
s= 1/2(13+14+15)
s= 21
Now Heron's formula for area, 'A':
A = sqrt(s(s-a)(s-b)(s-c))
A = sqrt(21(21-13)(21-14)(21-15))
A = sqrt(7056)
A = 84
With the area, 'A', and the height, 'h' we can calculate the base, 'h', using the well known formula A=1/2(bh), b=2A/h:
84 = 1/2(12b)
84 = 6b
84/6 = b
14 = b

Draw the triangle. Notice that the longest side 15 must overhang a base of 14 if the other side is 13. This implies that the base must be greater than 14. Can the shape of the triangle be determined mathematically? Yes:

We can use the Law of Cosines, c^2 =a^2 +b^2 -2abcos(C) where 'C' is the angle opposite the side 'c' and so on, to find the angles:

15^2 = 13^2+14^2-2(13)(14)cos(C)
225 = 169 + 196 - 182cos(C)
225 = 365 - 182cos(C)
-140 = -182cos(C)
cos(C) = 140/182
C = arccos(140/182)
C is approximately 39.7151 degrees

We need another angle to be able to determine all three angles, so we will calculate 'B' opposite 'b'.

b^2 = a^2 + c^2 - 2accos(B)
14^2 = 13^2 + 15^2 -2(13)(15)cos(B)
196 = 169 + 225 - 390cos(B)
196 = 394 - 390cos(B)
390cos(B) = 198
cos(B) = 198/390
B = arccos(198/390)
B is approximately: 59.4898 degrees

This forces A to be about 160.225 degrees

Notice, that this is curiously inconsistent. We know that the longest side of a triangle must be opposite its largest angle, however this is not the result we are obtaining.

Using this information we can verify are supposition that this is not a valid triangle based upon the curious result that the calculated base is less than the expected base. These incongruities reveal that there isn't a possible triangle with sides 13,14,15 and height 12. As such there is no solution to the problem.

What follows, my original response is the analysis of each possible triangle composed of sides, which are the subset of four consecutive integers that have a sum of 42. Unfortunately, I forgot to check whether these are valid triangles.

***

This is a deceptively complex problem that has many possibilities that have to be analyzed. The organization of steps here is a reflection of my thought process in solving the problem, so I apologize if its detail seems excessive.

First let’s find the four consecutive integers:
We'll start with the three sides of the triangle:
n+(n+1)+(n+2)= 42
3n+3=42
3n= 39
n=13

So the four integers can be either 13, 14, 15, 16 OR 12, 13, 14, 15.

Regardless of the four we choose, this is obviously a scalene triangle. Notice that we only have values for the sides and the height. WE DO NOT LET KNOW THE MEASURE OF THE BASE.

Before we can distinguish the sides from the height, we need to identify what three numbers give us a valid triangle. The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle is greater than the length of the third side. We will use this to identify what numbers are valid for the sides of the triangle:

Using 13,14,15,16.

13+14 > 16
13+16 > 14
14 +16 > 13
So 13,14, and 16 are an option, but so is 13, 14, 15:
13+14 > 15
13+15 >14
14+15 > 13
However the second case would force 16 to be the height of the triangle, which is impossible, for the height can not be greater than the greatest side. So we will use 13,14, and 16 for the triangles sides and suppose that 15 is the height.


Now we must check to see if 15 is indeed the height of this triangle: Counter-intuitively, to do this, we first need to find the area of the triangle. The area of the triangle can actually be found without knowing the base and height . We can use Heron's Formula to find the area of a triangle given only the measures of the three sides. Once we have found the area, we can see if 15 is a factor of twice the area: for a=bh/2.

Heron's Formula requires the calculation of the semi-perimeter(s). The semi-perimeter is the sum of the sides divided by 2.
a=13
b=14
c=16

s = 1/2(a+b+c)
s = 1/2(13+14+16)
s=43/2

Heron's Formula:
A = sqrt(s(s-a)(s-b)(s-c))
A=sqrt(43/2(43/2-13)(43/2-14)(43/2-16))
A=sqrt(120615/16)
A is approximately 86.8242
2A is approximately 173.648
Factoring 15: 11.5766
We can now tell that 15 is not the height of this triangle, because in a scalene triangle the top of the triangle overhangs the lowest side, and as such the base is slightly larger than the smallest size. Here the proposed base, 11.5766 is smaller than 13 and as such is invalid. Therefore we can tell that the four integers are not 13,14,15,16.

However we are not limited to 15 as the height, 13 or 14 could be the height, if we interpret 15 as a side. So we must also try those two scenarios. For brevity I will skip the calculations and merely report the outcomes, the process is the same as was outlined using 15 as the height:

*For 13in, we get an area of approximately 96.558 in^2 and a base of approximately 14.8551in, which is indeed larger than the smallest side, 14in. So it is a possibility that 13in is the height of the triangle and that the area is approximate 96.558 in^2.

*For 14in, we get an area of approximately 91.1921in^2 and a base of 13.0274in, which is indeed greater than 13in and therefore this is also a valid option.

We're not finished let, however, there are still the options associated with the other set of possible integers, 12, 13, 14, 15. It is trivial to see that valid triangles can be formed with any three of these numbers as the lengths of the sides. Furthermore we know that 15in cannot be the measure of the height, so we are left with three possible triangles: (12,13,15 ), (13,14,15), (12,14,15). We will test these possibilities using the same methods described above. Again for brevity, I'll skip directly to the conclusions:

For (12,13,15) with a height of 14:
area approximately: 74.8331
base approximately: 10.6904
Therefore this is not an option- the needed base is shorter than any of the sides.

*For (13,14,15) with a height of 12:
area: 84
base: 14
Therefore this is a valid triangle. The side of length 14 is the base.

*For (12,14,15) with a height of 13:
area approximately: 78.9268
base approximately: 12.1426
Thus, this is also an option.

***We now have four options:

A.) Sides: 13,14,15
height: 12
supposed base: 14
area: 84

B.) Sides: 12,14,15
height: 13
supposed base approximately: 12.1426
area approximately: 78.9268

C.) Sides: 14, 15, 16
height: 13
supposed base approximately: 14.8551
area approximately: 96.558

D.) sides: 13,15,16
height: 14
supposed base approximately: 13.0274
area approximately: 91.1921

Now we must see if we can narrow this done. To do this, I will check to see whether the supposed base is an actual possibility given the sides of the triangle.

Option (A)'s base is obviously possible.

The height of a scalene triangle is the length of a perpendicular dropped from the upper most vertex to the base. So the longest side forms a right triangle with the base and the height: the longest side is the hypotenuse. To verify whether the base is a possibility, we will analyze whether it satisfies the Pythagorean theorem:

B.) 15^2 = 13^2 + (12.1426)^2
225 = 169 + 147.443
225 != 316.443
Therefore option B is not an answer.

C.) 16^2 = 13^2 + (14.8551)^2
256 = 169 + 220.674
256 != 389.674
Therefore option C is not an answer.

D.) 16^2 = 14^2 +(13.0274)^2
256 = 196 + 169.713
256 != 365.713

***Now, finally we have determined all the options, and narrowed it down to only one possibility:

Sides: 13,14,15
height: 12
base: 14
area: 84
The numbers are 12,13,14,15 and the area is 84in^2

The selection of numbers from the two possible groups was not at all obvious, and the determination of the base and height was not trivial. To ensure the validity and uniqueness of the answer, all possibilities needed to be analyzed. I hope this helps.

Answer 3

Let's ignore the height, and take a look at the sides themselves.

We would obviously need to start with (x+1) because it is the second consecutive integer. The height would represent 'x'.

x+1+x+2+x+3 = 42
3x + 6 = 42
3x = 36
x = 12

The height is 12, so the base must be 15.

Let's check that to make sure

13 + 14 + 15 = 42 <- sides add up correctly.

Recall the formula for area of a triangle:

A = 1/2*b*h (where 'b' represents the base and 'h' represents the height)

Let's plug in our numbers now.

A = 1/2(15)*12
= 90 inches ^2

The answer is 90 inches ^2. Remember, area is always measured with the unit squared.

Good luck.

Answer 4

For a triangle of sides 13, 14, and 15 inches
with 15 inches as the base.

Using Herron's Formula
A = √(S(S - a)(S - b)(S - c))

where
S = (a + b+ c)/2

and
Height = (2/c)( (√(s(s-a)(s-b)(s-c))
where c is the base

I calculate 84 in² as the area

and

11.2 inches as the altitude of the triangle

For an area of 90 in² and an altitude of 12 inches,
the sides must be 14.306, 14, and 15 inches
.