# Math Help - Finding the area of a triangle

1. ## Finding the area of a triangle

I'm reviewing my most hated math: Geometry. I do not know how to get the answer to this question.

Triangle ABC is to be drawn so that the length of side AC is greater than 3 and less than 5. If the length of the perpendicular drawn from point B to side AC is less than 8, which of the following CANNOT be the area of the triangle.

(A) 8
(B) 10
(C) 16
(D) 18
(E) 36

2. Originally Posted by Mariolee
I'm reviewing my most hated math: Geometry. I do not know how to get the answer to this question.

Triangle ABC is to be drawn so that the length of side AC is greater than 3 and less than 5. If the length of the perpendicular drawn from point B to side AC is less than 8, which of the following CANNOT be the area of the triangle.

(A) 8
(B) 10
(C) 16
(D) 18
(E) 36
1. The area of a triangle is calculated by:

$area = \frac12 \cdot base \cdot height$

2. Taking the maximum dimension of the triangle you have
$base = 5$
$height = 8$

thus the maximum area would be $area = \frac12 \cdot 5 \cdot 8 = 20$

3. Now it's obvious which result can't be the area of the triangle.

3. Originally Posted by earboth
1. The area of a triangle is calculated by:

$area = \frac12 \cdot base \cdot height$

2. Taking the maximum dimension of the triangle you have
$base = 5$
$height = 8$

thus the maximum area would be $area = \frac12 \cdot 5 \cdot 8 = 20$

3. Now it's obvious which result can't be the area of the triangle.
Oh, so you just take the maximum. But wouldn't the maximum be like 4.99 multiplied by 7.99? :P Thank you!

4. No

The maximum area is 20 and the minimum area is $\frac12 \cdot 3 \cdot 8 = 12$

5. Originally Posted by Mariolee
... I do not know how to get the answer to this question.

Triangle ABC is to be drawn so that the length of side AC is greater than 3 and less than 5. If the length of the perpendicular drawn from point B to side AC is less than 8, which of the following CANNOT be the area of the triangle.

(A) 8
(B) 10
(C) 16
(D) 18
(E) 36
Originally Posted by running-gag
No

The maximum area is 20 and the minimum area is $\frac12 \cdot 3 \cdot 8 = 12$
I don't want to pick at you but as far as I understand the question the height of the triangle can range from zero to 8. Therefore the minimum area of the triangle can be zero.

6. Originally Posted by earboth
I don't want to pick at you but as far as I understand the question the height of the triangle can range from zero to 8. Therefore the minimum area of the triangle can be zero.
You are right !

7. Yea, the area can't be bigger than 20. They're right.

8. Originally Posted by Mariolee
I'm reviewing my most hated math: Geometry. I do not know how to get the answer to this question.

Triangle ABC is to be drawn so that the length of side AC is greater than 3 and less than 5. If the length of the perpendicular drawn from point B to side AC is less than 8, which of the following CANNOT be the area of the triangle.

(A) 8
(B) 10
(C) 16
(D) 18
(E) 36
Hi Mariolee,

you can stand the triangle on "any" of it's 3 sides.

The area of the triangle is the same no matter which side we stand it on.
suppose we stand it on side AC.
Then point B will be the topmost point and is less than 8 units above AC.
So the perpendicular height is less than 8.
The base AC is between 3 and 5 units long.

A triangle is half of a parallelogram.
a parallelogram has the same area as a rectangle of same base and perpendicular height.
The rectangle's area is (base)(height).
the triangle's area is half that.

We don't know the height of the triangle, only that it cannot be 8 or more.
Hence the triangle's area cannot reach 0.5(5)8 square units.