Geometry - How to find the length of the diagonal of a kite

A kite has an area of square units, and one diagonal is units longer than the other. In unites, what is the length of the shorter diagonal?

Explanation

Let be the length of the shorter diagonal. Then the length of the longer diagonal can be represented by .

Recall how to find the area of a kite:

$\text{Area}=\frac{(\text{Diagonal 1})(\text{Diagonal 2})}{2}$

Plug in the given area and solve for .

$119=\frac{x(x+3)}{2}$

Since we are dealing with geometric shapes, the answer must be a positive value. Thus, .

The length of the shorter diagonal is units long.

If the length of the shorter diagonal is four, what is the length of the longer diagonal of this kite?

Kite 1

$4\sqrt5$

$\sqrt{45}$

$\sqrt5$

$3\sqrt5$

Explanation

We can find the longer diagonal by adding together the altitude of the top triangle and the altitude of the bottom triangle. To find these, use Pythagorean Theorem. We can use Pythagorean Theorem because one of the properties of a kite is that the two diagonals are perpendicular.

The top triangle has two sides of length 3 \[labeled in the picture\], and a base of 4 \[provided in the written directions\]. To figure out the altitude, split this triangle into 2 right triangles. The two legs are x \[the altitude\] and 2 \[half of the base 4\], and the hypotenuse is 3:

subtract 4 from both sides

take the square root of both sides

$x=\sqrt5$

We will do something similar for the bottom triangle. Consider one of the right triangles. It will have a hypotenuse of 7, one leg that we don't know, x \[the altitude\], and one leg 2 \[half the shorter diagonal\]. Set up the equation using the Pythagorean Theorem:

subtract 4 from both sides

take the square root of both sides

$x = \sqrt{45}$

That can be simplified by considering 45 as the product of . Since the square root of 9 is 3, we can re-write $\sqrt{45}$ as $3\sqrt{5}$ .

Adding together the first answer of $\sqrt{5}$ plus $3\sqrt5$ gives $4\sqrt5$ .

If the area of a kite is square units, and one diagonal is units longer than the other, what is the length of the shorter diagonal?

Explanation

Let be the length of the shorter diagonal. Then the length of the longer diagonal can be represented by .

Recall how to find the area of a kite:

$\text{Area}=\frac{(\text{Diagonal 1})(\text{Diagonal 2})}{2}$

Plug in the given area and solve for .

$198=\frac{x(x+4)}{2}$

Since we are dealing with geometric shapes, the answer must be a positive value. Thus, .

The length of the shorter diagonal is units long.

If the area of a kite is square units, and one diagonal is units longer than the other, what is the length of the longer diagonal?

Explanation

Let be the length of the shorter diagonal. Then the length of the longer diagonal can be represented by .

Recall how to find the area of a kite:

$\text{Area}=\frac{(\text{Diagonal 1})(\text{Diagonal 2})}{2}$

Plug in the given area and solve for .

$20=\frac{x(x+3)}{2}$

Since we are dealing with geometric shapes, the answer must be a positive value. Thus, .

To find the longer diagonal, add .

The length of the longer diagonal is units long.

A kite has two perpendicular interior diagonals. One diagonal has a measurement of and the area of the kite is $45\textup{ units}^{2}$ . Find the length of the other interior diagonal.

Explanation

This problem can be solved by applying the area formula:

Since this question provides the area of the kite and length of one diagonal, plug that information into the equation to solve for the missing diagonal.

Thus the solution is:

$45=\frac{18\times diagonal B}{2}$

$45\times2=18\times diagonalB$

$diagonal B=\frac{90}{18}=5$

A kite has two perpendicular interior diagonals. One diagonal has a measurement of and the area of the kite is $54\textup{ units}^{2}$ . Find the length of the other interior diagonal.

Explanation

This problem can be solved by applying the area formula:

Since this question provides the area of the kite and length of one diagonal, plug that information into the equation to solve for the missing diagonal.

Thus the solution is:

$54=\frac{9\times diagonal B}{2}$

$54\times2=9\times diagonalB$

$diagonal B=\frac{108}{9}=12$

If the area of a kite is square units, and the length of one diagonal is units shorter than the other, what is the length of the shorter diagonal?

Explanation

Let be the length of the shorter diagonal. Then the length of the longer diagonal can be represented by .

Recall how to find the area of a kite:

$\text{Area}=\frac{(\text{Diagonal 1})(\text{Diagonal 2})}{2}$

Plug in the given area and solve for .

$1680=\frac{x(x+4)}{2}$

Since we are dealing with geometric shapes, the answer must be a positive value. Thus, .

The length of the shorter diagonal is units long.

A kite has two perpendicular interior diagonals. One diagonal has a measurement of and the area of the kite is $6\textup{,}250\textup{ units}^{2}$ . Find the sum of the two perpendicular interior diagonals.

$2\textup{,}000$

Explanation

First find the length of the missing diagonal before you can find the sum of the two perpendicular diagonals.

To find the missing diagonal, apply the area formula:

This question provides the area of the kite and length of one diagonal, plug that information into the equation to solve for the missing diagonal.

$6,250=\frac{250\times diagonal B}{2}$

$6250\times2=250\times diagonalB$

$diagonal B=\frac{12,500}{250}=50$

Therefore, the sum of the two diagonals is:

A kite has two perpendicular interior diagonals. One diagonal has a measurement of $22\textup{mm}$ and the area of the kite is $297\textup{mm}^{2}$ . Find the length of the other interior diagonal.

$27\textup{mm}$

$25\textup{mm}$

$37\textup{mm}$

$35\textup{mm}$

$19\textup{mm}$

Explanation

This problem can be solved by applying the area formula:

Since this question provides the area of the kite and length of one diagonal, plug that information into the equation to solve for the missing diagonal.

Thus the solution is:

$297=\frac{22\times diagonal B}{2}$

$297\times2=22\times diagonalB$

$diagonal B=\frac{594}{22}=27$

If the area of the kite is square units, and the difference between the lengths of the diagonals is units, what is the length of the shorter diagonal?