Two sides of a triangle are 4 and 5. Then, for the area of the triangle, which one of the following bounds is the sharpest?a)<10.b)≤10.c)< 8.d)> 5.Correct answer is option 'B'. Can you explain this answer?

Question

Two sides of a triangle are 4 and 5. Then, for the area of the triangle, which one of the following bounds is the sharpest?a)&#60;10.b)&#8804;10.c)&#60; 8.d)&#62; 5.Correct answer is option 'B'. Can you explain this answer?

Answer

9
b) 10
c) 11
d) 12
e) 13

The sharpest bound can be found using the formula for the area of a triangle: A = (1/2)bh, where b is the base and h is the height. We know the lengths of two sides of the triangle, but we need to find the height in order to calculate the area.

Using the Pythagorean theorem, we can find the length of the third side of the triangle: c^2 = a^2 + b^2, where c is the hypotenuse (the unknown side), and a and b are the known sides. Plugging in a = 4 and b = 5, we get:

c^2 = 4^2 + 5^2
c^2 = 41
c ≈ 6.4

Now we can use the formula for the area of a triangle: A = (1/2)bh, where b = 4 and h is the height.

We can use the Pythagorean theorem again to find the height: h^2 = c^2 - b^2. Plugging in c ≈ 6.4 and b = 4, we get:

h^2 = 6.4^2 - 4^2
h^2 ≈ 18.56
h ≈ 4.3

So the area of the triangle is:

A = (1/2)(4)(4.3)
A ≈ 8.6

The closest bound is 9, so the answer is (a) 9.

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