A kite with a 10cm height and a 5 cm width.
1. By what factor does the area change if both diagonals are doubled? Explain.
2. By what factor does the area change if one diagonal is doubled? Explain.
1) Area of the kite = (p*q)/2
Where p and q are both the diagonals.
A(initial) = (p*q)/2
A(final) = ((2p)*(2q))/2
= 4(p*q)/2
= 4*A(inital)
It increases by a factor of 4.
2)
A(initial) = (p*q)/2
A(final) = ((2p)*(q))/2
= 2(p*q)/2
= 2*A(inital)
It increases by a factor of 2.
To answer both of these questions, we need to understand the relationship between the diagonals of a kite and its area. Let's start with some background information.
In a kite, the diagonals are line segments that connect opposite vertices. The longer diagonal is the one that bisects the shorter diagonal, creating four right angles at the intersection. The area of a kite can be calculated by multiplying half the product of the diagonals.
Now let's move on to the first question.
1. By what factor does the area change if both diagonals are doubled?
To solve this, we need to find the original area of the kite. The area of a kite is given by the formula (1/2) * d1 * d2, where d1 and d2 are the lengths of the diagonals.
In this case, let's assume the original lengths of the diagonals are d1 = x and d2 = y. Therefore, the original area of the kite is (1/2) * x * y.
If we double both diagonals, the new lengths would be 2x and 2y. So the new area of the kite would be (1/2) * 2x * 2y = 2 * (1/2) * x * y = 2 times the original area.
Therefore, if both diagonals of the kite are doubled, the area will increase by a factor of 2.
Now let's move on to the second question.
2. By what factor does the area change if one diagonal is doubled?
To solve this, let's once again assume the original lengths of the diagonals as d1 = x and d2 = y.
If one diagonal is doubled, let's say d1 becomes 2x while d2 remains y. The new area of the kite would be (1/2) * 2x * y.
Dividing the new area by the original area, we get ((1/2) * 2x * y) / ((1/2) * x * y) = (2x * y) / (x * y) = 2.
Therefore, if one diagonal of the kite is doubled, the area will increase by a factor of 2.
In summary:
1. If both diagonals are doubled, the area will increase by a factor of 2.
2. If one diagonal is doubled, the area will also increase by a factor of 2.