How Would i do this?

Question

One side of a triangle has length 8 and a second side has length 5. Which of the following could be the area of the triangle?


I 24
II 20
III 5

A. I only
B. II only
C. III only
D. II and III only
E. I, II and III

The correct answer is D

Answer 1

it's true that the area of a triangle is 1/2 x base x height.

but without knowing the third side, you can't know both the base and the height. only one or the other + the length of the second side.

examples

triangle with angle Θ between base of 8 and leg of 5 units...

picture it like this.

o-------- with a second line starting at
"o" and going up and to the right 5 units. the connect the end of that line to the end of the horizontal line. the angle Θ (the angle at "o") can vary fyi. then drop a line from the peak straight down to the horizontal line. that's the height. if the base is "8" and the leg rising up and to the right is "5", the height is not "5". the height / the hypotenus = sin Θ. and the hypotenus is "5"

let's say the angle is 10°

sin 10° = h/5. so h = .86 and 1/2 base x height = 3.47 square units.

if the angle is 60 °, h = 4.33, and area = 17.36

******

if area = 5, then 1/2 x base x height = 5 , --> height = 10/base = 10/8

and angle between sides can be found by

sin Θ = (10/8) / 5 = 2/8 = 0.25

so Θ = sin^-1 (.25) = 14.47°

*******

if area = 20,

height =40 / base = 40/8 = 5

Θ = sin^-1 (5/5) = 90°

*****

if area = 24,

height = 48 / base = 48/8 = 6

and

Θ = sin^-1 (6/5) = impossible


so possible areas are 20, and 4 which means "D" is the correct answer.

Answer 2

Well we know that the area is 1/2 bh

So if the triangle has the 8 side as the base = 1/2 8h or 4h. In this case, if the height was = to the other side if woud be a total area of 20. SO we know II is a valid answer and A, C are out.

Ok, now, what if the height is different from the other side (which it could be). For these cases, the maximum the height could be is 5. So far so good.

Ok = if the triangle has the 5 side as the base = 1/2 (5)h or 2.5 h. In this case, if the height was = to the other side if woud be a total area of 20, just like before. But what if the height is different from the other side (which it could be). For these cases, the maximum the height could be is 8. So the maximum area is 20 for all cases. So E is out.

That leaves us with B and D as only possible answers. The question is, if the base is the 5 side at an angle such that the height was less than 2, we could have 1/2 (5)(2) which could equal 5. This situation could exist as a possibility. So D is the correct answer.

Just to clarify, if the 8 side was at an angle to the 5 side such that the corresponding height was 2, you could have an area of 5. Hope this helps.

Answer 3

One of the formulas for the area of a triangle is absin(C)/2 where a and b are side lengths and C is the angle between the a and b. To prove this draw a triangle and labeling two sides a and b and the angle in between them C. Then draw the altitude (height) to side a. You can use trig to express height in terms of side b and angle C. Then you just substitute into 1/2 base*height to obtain that formula.

In this case the area (A) would be

(8)(5)sin(C)/2
=20sin(C).

In a triangle 0 < C < 180 degrees. That means that 0 < sinC <=1. As such our area A varies from 0 < A <=20. In sum, our area can be any positive real number less than or equal to 20. Thus the area could be II (20) or III (5) but not I (24), so our answer is D (II and III only).

Answer 4

Since the area is (1/2)bh, this means bh = 48, 40, or 10.

Treat 8 as the base. Then the height, relative to that side, must be 6, 5 or 10/8=1.25.

It's possible that the 5 side is also the height, in which case you have a right triangle. But you can't have a height higher than that or else you wouldn't be able to connect the triangle. To show this, draw the base as "8", then connect the "5" segment to one end. You can open or close the angle of these two to any degree, but the height (perpendicular line drawn from the tip of the "5" side to the "8" base) is never above 5. So the answer is D.

Answer 5

D.

Imagine the long side down flat and imagine varying the angle of the short side to make all possible triangles. The minimum area is practically zero when the two sides are almost parallel. The largest area happens when the sides are perpendicular and equals 5x8/2 = 20.

Answer 6

area of triangle = 1/2 base * height
... If you say 1/2 B* H or 1/2 H *B the answer will always be the same
1/2 of 8 * 5 = 20
Answer is B. (II only)

Answer 7

b because area is 1/2 times base times height = 20

Answer 8

B. 20 only
They would have to be the legs of a rjght triangle.
So 8*5/2 = 20

Answer 9

think of a triangle as half of a square, if one side is 8 and one side is 5...then the area must be 20..
(8x5)/2. :) hope i helped

Answer 10

B